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In which are explained 
The causes of that which occurs 


And particularly 

In the strange REFRACTION 



Rendered into English 





WROTE this Treatise during my sojourn 
in France twelve years ago, and I com- 
municated it in the year 1 678 to the learned 
persons who then composed the Royal 
Academy of Science, to the membership 
of which the King had done me the honour 
of calling me. Several of that body who are still alive will 
remember having been present when I read it, and above 
the rest those amongst them who applied themselves par- 
ticularly to the study of Mathematics ; of whom I cannot 
cite more than the celebrated gentlemen Cassini, Romer, 
and De la Hire. And although I have since corrected and 
changed some parts, the copies which I had made of it at 
that time may serve for proof that I have yet added no- 
thing to it save some conjectures touching the formation of 
Iceland Crystal, and a novel observation on the refraction 
of Rock Crystal. I have desired to relate these particulars 
to make known how long I have meditated the things 
which now I publish, and not for the purpose of detraCt- 
ing from the merit of those who, without having seen any- 
thing that I have written, may be found to have treated 



of like matters : as has in fa<5t occurred to two eminent 
Geometricians, Messieurs Newton and Leibnitz, with re- 
spe6l to the Problem of the figure of glasses for collecting 
rays when one of the surfaces is given. 

One may ask why I have so long delayed to bring this 
work to the light. The reason is that I wrote it rather 
carelessly in the Language in which it appears, with the 
intention of translating it into Latin, so doing in order to 
obtain greater attention to the thing. After which I pro- 
posed to myself to give it out along with another Treatise 
on Dioptrics, in which I explain the effeds of Telescopes 
and those things which belong more to that Science. But 
the pleasure of novelty being past, I have put off from 
time to time the execution of this design, and I know not 
when I shall ever come to an end if it, being often turned 
aside either by business or by some new study. Consider- 
ing which I have finally judged that it was better worth 
while to publish this writing, such as it is, than to let it 
run the risk, by waiting longer, of remaining lost. 

There will be seen in it demonstrations of those kinds 
which do not produce as great a certitude as those of 
Geometry, and which even differ much therefrom, since 
whereas the Geometers prove their Propositions by fixed 
and incontestable Principles, here the Principles are veri- 
fied by the conclusions to be drawn from them; the nature 
of these things not allowing of this being done otherwise. 
It is always possible to attain thereby to a degree of prob- 
ability which very often is scarcely less than complete proof. 
To wit, when things which have been demonstrated by the 
Principles that have been assumed correspond perfectly to 
the phenomena which experiment has brought under ob- 
servation ; especially when there are a great number of 



them, and further, principally, when one can imagine and 
foresee new phenomena which ought to follow from the 
hypotheses which one employs, and when one finds that 
therein the fad: corresponds to our prevision. But if all 
these proofs of probability are met with in that which I 
propose to discuss, as it seems to me they are, this ought 
to be a very strong confirmation of the success of my in- 
quiry; and it must be ill if the fa<5ts are not pretty much 
as I represent them. I would believe then that those who 
love to know the Causes of things and who are able to 
admire the marvels of Light, will find some satisfaction in 
these various speculations regarding it, and in the new 
explanation of its famous property which is the main 
foundation of the construction of our eyes and of those 
great inventions which extend so vastly the use of them. 
I hope also that there will be some who by following these 
beginnings will penetrate much further into this question 
than I have been able to do, since the subject must be far 
from being exhausted. This appears from the passages 
which I have indicated where I leave certain difficulties 
without having resolved them, and still more from matters 
which I have not touched at all, such as Luminous Bodies 
of several sorts, and all that concerns Colours; in which 
no one until now can boast of having succeeded. Finally, 
there remains much more to be investigated touching the 
nature of Light which I do not pretend to have disclosed, 
and I shall owe much in return to him who shall be able 
to supplement that which is here lacking to me in know- 
ledge. The Hague. The 8 January 1690. 


ONSIDERING the great influence which 
this Treatise has exercised in the develop- 
ment of the Science of Optics, it seems 
strange that two centuries should have 
passed before an English edition of the 
work appeared. Perhaps the circumstance 
is due to the mistaken zeal with which formerly every- 
thing that conflicted with the cherished ideas of Newton 
was denounced by his followers. The Treatise on Light 
of Huygens has, however, withstood the test of time: and 
even now the exquisite skill with which he applied his 
conception of the propagation of waves of light to unravel 
the intricacies of the phenomena of the double refraftion 
of crystals, and of the refra6tion of the atmosphere, will 
excite the admiration of the student of Optics. It is true 
that his wave theory was far from the complete doctrine 
as subsequently developed by Thomas Young and Augustin 
Fresnel, and belonged rather to geometrical than to physical 
Optics. If Huygens had no conception of transverse vibra- 
tions, of the principle of interference, or of the existence 
of the ordered sequence of waves in trains, he nevertheless 
attained to a remarkably clear understanding of the prin- 

b ciples 


ciples of wave-propagation; and his exposition of the sub- 
jeCt marks an epoch in the treatment of Optical problems. 
It has been needful in preparing this translation to exer- 
cise care lest one should import into the author's text 
ideas of subsequent date, by using words that have come 
to imply modern conceptions. Hence the adoption of as 
literal a rendering as possible. A few of the author's terms 
need explanation, He uses the word " refraction," for 
example, both for the phenomenon or process usually so 
denoted, and for the result of that process: thus the re- 
fraCted ray he habitually terms " the refraction " of the 
incident ray. When a wave-front, or, as he terms it, a 
u wave," has passed from some initial position to a subse- 
quent one, he terms the wave-front in its subsequent 
position " the continuation " of the wave. He also speaks 
of the envelope of a set of elementary waves, formed by 
coalescence of those elementary wave-fronts, as " the ter- 
mination" of the wave; and the elementary wave-fronts 
he terms " particular " waves. Owing to the circumstance 
that the French word rayon possesses the double signifi- 
cation of ray of light and radius of a circle, he avoids its 
use in the latter sense and speaks always of the semi- 
diameter, not of the radius. His speculations as to the 
ether, his suggestive views of the structure of crystalline 
bodies, and his explanation of opacity, slight as they are, 
will possibly surprise the reader by their seeming modern- 
ness. And none can read his investigation of the phenomena 
found in Iceland spar without marvelling at his insight 
and sagacity. 

S. P. T. 

June, 1912. 


Contained in this 'Treatise 

CHAP. I. On Rays Propagated in 
Straight Lines. 

That Light is produced by a certain 
movement. p. 3 

That no substance passes from the lumin- 
ous objeft to the eyes. p. 3 

That Light spreads spheric ally ^ almost 
as Sound does. p. 4 

Whether Light takes time to spread. 


Experience seeming to prove that it passes 

instantaneously. p. 5 

Experience proving that it takes time. 

p. 8 
How much its speed is greater than that 

of Sound. p. 10 

In what the emission of Light differs 

from that of Sound. p. 10 

That it is not the same medium which 

serves for Light and Sound, p. 1 1 
How Sound is propagated. p. 12 

How Light is propagated. p. 14 

Detailed Remarks on the propagation 

of Light. p. 15 

Why Rays are propagated only in 

straight lines. p. 2O 

How Light coming in different directions 

can cross itself. p. 22 

CHAP. II. On Reflexion. 

Demonstration of equality of angles of 
incidence and reflexion. p. 23 

Why the incident and reflected rays are 
in the same plane perpendicular to the 
reflecting surface. p. 25 

That it is not needful for the reflecting 
surface to be perfectly flat to attain 
equality of the angles of incidence and 
reflexion. p. 27 

CHAP. III. On Refradlion. 

That bodies may be transparent without 
any substance passing through them. 

p. 29 

Proof that the ethereal matter passes 
through transparent bodies. p. 30 

How this matter passing through can 
render them transparent. p. 31 

That the most solid bodies in appearance 
are of a very loose texture. p. 31 

That Light spreads more slowly in water 
and in glass than in air. p. 32 

Third hypothesis to explain transpar- 
ency^ and the retardation which Light 
suffers. .A3 2 

On that which makes bodies opaque. 

P- 34 

Demonstration why Refraction obeys the 
known proportion of Sines. p. 35 

Why the incident and refrafted Rays 
produce one another reciprocally, p. 39 

Why Reflexion within a triangular 
glass prism is suddenly augmented 
when the Light can no longer pene- 
trate, p. 40 

That bodies which cause greater Refrac- 
tion also cause stronger Reflexion. 

p. 42 

Demonstration of the Theorem of Mr. 
Fermat. p. 43 

CHAP. IV. On the Refradion of 
the Air. 

That the emanations of Light in the air 
are not spherical. p t 45 

How consequently some objects appear 
higher than they are. p. 47 

How the Sun may appear on the Hori- 
zon before he has risen. p. 49 


That the rays of light become curved in 
the Air of the Atmosphere, and what 
effects this produces. p. 50 

CHAP. V. On the Strange Refrac- 
tion of Iceland Crystal. 
That this Crystal grows also in other 

countries. p. 5 2 

Wlw first wrote about it. ^'53 

Description of Iceland Crystal; its sub- 

stance, shape, and properties, p. 53 
That it has two different Refractions. 

P- 54 

That the ray perpendicular to the surface 
suffers refraction, and that some rays 
inclined to the surface pass without 
suffering refraction. p. 55 

Observation of the refractions in this 
Crystal. p. 56 

That there is a Regular and an Ir- 
regular Refraction. P- 57 

The way of measuring the two Refrac- 
tions of Iceland Crystal. P* 57 

Remarkable properties of the Irregular 
Refraction. p. 60 

Hypothesis to explain the double Refrac- 
tion, p. 6 1 

That Rock Crystal has also a double 
Refraction, p. 62 

Hypothesis of emanations ofLight, with- 
in Iceland Crystal, of spheroidal form^ 
for the Irregular Refraction, p. 63 

How a perpendicular ray can suffer 
Refraction. p. 64 

How the position and form of the spher- 
oidal emanations in this Crystal can 
be defined. p. 65 

Explanation of the Irregular Refraction 
by these spheroidal emanations, p. 67 

Easy way to find the Irregular Re- 
fraction of each incident ray. p. 70 

Demonstration of the oblique ray which 


traverses the Crystal without being 
refracted. p. 73 

Other irregularities of Refraction ex- 
plained. />. 76 

That an obj eft placed beneath the Crystal 
appears double^in two images of differ- 
ent heights, p. Si 

Why the apparent heights of one of the 
images change on changing the position 
of the eyes above the Crystal, p. 85 

Of the different sections of this Crystal 
which produce yet other refractions^ 
and confirm all this Theory, p. 88 

Particular way of polishing the surfaces 
after it has been cut. p. 9 1 

Surpr is ing phenomenon touching the rays 
which pass through two separated 
pieces; the cause of which is not ex- 
plained, p. 92 

Probable conjecture on the internal com- 
position of Iceland Crystal^ and of 
what figure its particles are. p. 95 

Tests to confirm this conjecture, p. 97 

Calculations which have been supposed 
in this Chapter. p. 99 

CHAP. VI. On the Figures of 
transparent bodies which serve 
for Refraction and for Reflexion. 

General and easy rule to find these 
Figures. p. 106 

Invention of the Ovals of Mr. Des 
Cartes for Dioptrics. p. 109 

How he was able to find these Lines. 

p. 114 

{lass for 

Way of finding the surface of a git 
perfeft refracJion, when the other 
surface is given. p. 116 

Remark on what happens to rays re- 
frafted at a spherical surface, p. 123 

Remark on the curved line which is 
formed by reflexion in a spherical con- 
cave mirror. p. 1 26 



S happens in all the sciences in which 
Geometry is applied to matter, the demon- 
strations concerning Optics are founded 
on truths drawn from experience. Such 
are that the rays of light are propagated 
in straight lines; that the angles of re- 
flexion and of incidence are equal; and that in refraftion 
the ray is bent according to the law of sines, now so well 
known, and which is no less certain than the preceding 

The majority of those who have written touching the 
various parts of Optics have contented themselves with 
presuming these truths. But some, more inquiring, have 
desired to investigate the origin and the causes, considering 
these to be in themselves wonderful effe<5ls of Nature. In 
which they advanced some ingenious things, but not 
however such that the most intelligent folk do not wish 
for better and more satisfactory explanations. Wherefore 
I here desire to propound what I have meditated on thesub- 


jet, so as to contribute as much as I can to the explanation 
of this department of Natural Science, which, not without 
reason, is reputed to be one of its most difficult parts. I 
recognize myself to be much indebted to those who were 
the first to begin to dissipate the strange obscurity in which 
these things were enveloped, and to give us hope that they 
might be explained by intelligible reasoning. But, on the 
other hand I am astonished also that even here these have 
often been willing to offer, as assured and demonstrative, 
reasonings which were far from conclusive. For I do not 
find that any one has yet given a probable explanation of the 
first and most notable phenomena of light, namely why it is 
not propagated except in straight lines, and how visible 
rays, coming from an infinitude of diverse places, cross one 
another without hindering one another in any way. 

I shall therefore essay in this book, to give, in accordance 
with the principles accepted in the Philosophy of the pre- 
sent day, some clearer and more probable reasons, firstly of 
these properties of light propagated re&ilinearly ; secondly 
of light which is reflected on meeting other bodies. Then 
I shall explain the phenomena of those rays which are said 
to suffer refraftion on passing through transparent bodies 
of different sorts; and in this part I shall also explain the 
effects of the refradlion of the air by the different densities 
of the Atmosphere. 

Thereafter I shall examine the causes of the strange re- 
fra6tion of a certain kind of Crystal which is brought from 
Iceland. And finally I shall treat of the various shapes of 
transparent and refle6ting bodies by which rays are collected 
at a point or are turned aside in various ways. From 
this it will be seen with what facility, following our new 
Theory, we find not only the Ellipses, Hyperbolas, and 



other curves which Mr. Des Cartes has ingeniously in- 
vented for this purpose; but also those which the surface 
of a glass lens ought to possess when its other surface is 
given as spherical or plane, or of any other figure that 
may be. 

It is inconceivable to doubt that light consists in the 
motion of some sort of matter. For whether one considers 
its produ<5tion, one sees that here upon the Earth it is 
chiefly engendered by fire and flame which contain with- 
out doubt bodies that are in rapid motion, since they 
dissolve and melt many other bodies, even the most solid ; 
or whether one considers its effects, one sees that when 
light is collected, as by concave mirrors, it has the property 
of burning* as a fire does, that is to say it disunites the particles 
of bodies. This is assuredly the mark of motion, at least 
in the true Philosophy, in which one conceives the causes 
of all natural effe6ls in terms of mechanical motions. This, 
in my opinion, we must necessarily do, or else renounce 
all hopes of ever comprehending anything in Physics. 

And as, according to this Philosophy, one holds as cer- 
tain that the sensation of sight is excited only by the 
impression of some movement of a kind of matter which 
a6ls on the nerves at the back of our eyes, there is here 
yet one reason more for believing that light consists in a 
movement of the matter which exists between us and the 
luminous body. 

Further, when one considers the extreme speed with 
which light spreads on every side, and how, when it comes 
from different regions, even from those diredtly opposite, 
the rays traverse one another without hindrance, one may 
well understand that when we see aluminous object, it cannot 
be by any transport of matter coming to us from this objeft, 



in the way in which a shot or an arrow traverses the air ; 
for assuredly that would too greatly impugn these two 
properties of light, especially the second of them. It is 
then in some other way that light spreads ; and that which 
can lead us to comprehend it is the knowledge which we 
have of the spreading of Sound in the air. 

We know that by means of the air, which is an invisible 
and impalpable body, Sound spreads around the spot where 
it has been produced, by a movement which is passed on 
successively from one part of the air to another ; and that 
the spreading of this movement, taking place equally 
rapidly on all sides, ought to form spherical surfaces ever 
enlarging and which 'strike our ears. Now there is no 
doubt at all that light also comes from the luminous body 
to our eyes by some movement impressed on the matter 
which is between the two; since, as we have already seen, it 
cannot be by the transport of a body which passes from 
one to the other. If, in addition, light takes time for its 
passage which we are now going to examine it will 
follow that this movement, impressed on the intervening 
matter, is successive ; and consequently it spreads, as Sound 
does, by spherical surfaces and waves: for I call them 
waves from their resemblance to those which are seen to 
be formed in water when a stone is thrown into it, and 
which present a successive spreading as circles, though 
these arise from another cause, and are only in a flat surface. 

To see then whether the spreading of light takes time, 
let us consider first whether there are any fadls of experience 
which can convince us to the contrary. As to those which 
can be made here on the Earth, by striking lights at great 
distances, although they prove that light takes no sensible 
time to pass over these distances, one may say with good 



reason that they are too small, and that the only conclusion 
to be drawn from them is that the passage of light is ex- 
tremely rapid. Mr. Des Cartes, who was of opinion that 
it is instantaneous, founded his views, not without reason, 
upon a better basis of experience, drawn from the Eclipses 
of the Moon; which, nevertheless, as I shall show, is not 
at all convincing. I will set it forth, in a way a little 
different from his, in order to make the conclusion more 

Let A be the place of the sun, BD a part of the orbit or 

annual path of the Earth: ABC a straight line which I 
suppose to meet the orbit of the Moon, which is represented 
by the circle CD, at C. 

Now if light requires time, for example one hour, to 
traverse the space which is between the Earth and the 
Moon, it will follow that the Earth having arrived at B, 
the shadow which it casts, or the interruption of the light, 
will not yet have arrived at the point C, but will only 
arrive there an hour after. It will then be one hour after, 
reckoning from the moment when the Earth was at B, 



that the Moon, arriving at C, will be obscured : but this 
obscuration or interruption of the light will not reach the 
Earth till after another hour. Let us suppose that the Earth 
in these two hours will have arrived at E. The Earth 
then, being at E, will see the Eclipsed Moon at C, which 
it left an hour before, and at the same time will see the 
sun at A. For it being immovable, as I suppose with 
Copernicus, and the light moving always in straight lines, 
it must always appear where it is. But one has always 
observed, we are told, that the eclipsed Moon appears at 
the point of the Ecliptic opposite to the Sun; and yet here it 
would appear in arrear of that point by an amount equal to 
the angle GEC, the supplement of AEC. This, however, 
is contrary to experience, since the angle GEC would be 
very sensible, and about 33 degrees. Now according to 
our computation, which is given in the Treatise on the 
causes of the phenomena of Saturn, the distance B A be- 
tween the Earth and the Sun is about twelve thousand 
diameters of the Earth, and hence four hundred times 
greater than BC the distance of the Moon, which is 30 
diameters. Then the angle ECB will be nearly four 
hundred times greater than BAE, which is five minutes; 
namely, the path which the earth travels in two hours 
along its orbit; and thus the angle BCE will be nearly 33 
degrees; and likewise the angle CEG, which is greater by 
five minutes. 

But it must be noted that the speed of light in this 
argument has been assufned such that it takes a time of 
one hour to make the passage from here to the Moon. If 
one supposes that for this it requires only one minute of 
time, then it is manifest that the angle CEG will only be 
33 minutes; and if it requires only ten seconds of time, 



the angle will be less than six minutes. And then it will not 
be easy to perceive anything of it in observations of the 
Eclipse; nor, consequently, will it be permissible to deduce 
from it that the movement of light is instantaneous. 

It is true that we are here supposing a strange velocity 
that would be a hundred thousand times greater than that 
of Sound. For Sound, according to what I have observed, 
travels about 180 Toises in the time of one Second, or in 
about one beat of the pulse. But this supposition ought 
not to seem to be an impossibility; since it is not a question 
of the transport of a body with so great a speed, but of a 
successive movement which is passed on from some bodies 
to others. I have then made no difficulty, in meditating 
on these things, in supposing that the emanation of light 
is accomplished with time, seeing that in this way all its 
phenomena can be explained, and that in following the 
contrary opinion everything is incomprehensible. For it 
has always seemed to me that even Mr. Des Cartes, whose 
aim has been to treat all the subje6ts of Physics intelligibly, 
and who assuredly has succeeded in this better than any one 
before him, has said nothing that is not full of difficulties, 
or even inconceivable, in dealing with Light and its pro- 

But that which I employed only as a hypothesis, has 
recently received great seemingness as an established truth 
by the ingenious proof of Mr. Romer which I am going 
here to relate, expecting him himself to give all that is 
needed for its confirmation. It is founded as is the preceding 
argument upon celestial observations, and proves not only 
that Light takes time for its passage, but also demonstrates 
how much time it takes, and that its velocity is even at 
least six times greater than that which I have just stated. 




For this he makes use of the Eclipses suffered by the 
little planets which revolve around Jupiter, and which 
often enter his shadow : and see what is his reasoning. 
Let A be the Sun, BCDE the annual orbit of the Earth, 
F Jupiter, GN the orbit of the nearest of his Satellites, for it 

is this one which is more apt for this 
investigation than any of the other 
three, because of the quickness of its 
revolution. Let G be this Satellite 
entering into the shadow of Jupiter, 
H the same Satellite emerging from 
the shadow. 

Let it be then supposed, the Earth 
being at B some time before the 
last quadrature, that one has seen 
the said Satellite emerge from the 
shadow; it must needs be, if the 
Earth remains at the same place, 
that, after 42^ hours, one would 
again see a similar emergence, be- 
cause that is the time in which it 
makes the round of its orbit, and 
when it would come again into op- 
position to the Sun. And if the 
Earth, for instance, were to remain 
always at B during 30 revolutions of this Satellite, one 
would see it again emerge from the shadow after 30 times 
42-^ hours. But the Earth having been carried along during 
this time to C, increasing thus its distance from Jupiter, 
it follows that if Light requires time for its passage the 
illumination of the little planet will be perceived later at 



C than it would have been at B, and that there must be 
added to this time of 30 times 42^ hours that which 
the Light has required to traverse the space MC, the 
difference of the spaces CH, BH. , Similarly at the other 
quadrature when the earth has come to E from D while 
approaching toward Jupiter, the immersions of the Satellite 
ought to be observed at E earlier than they would have 
been seen if the Earth had remained at D. 

Now in quantities of observations of these Eclipses, made 
during ten consecutive years, these differences have been 
found to be very considerable, such as ten minutes and 
more; and from them it has been concluded that in order 
to traverse the whole diameter of the annual orbit KL, 
which is double the distance from here to the sun, Light 
requires about 22 minutes of time. 

The movement of Jupiter in his orbit while the Earth 
passed from B to C, or from D to E, is included in this 
calculation; and this makes it evident that one cannot 
attribute the retardation of these illuminations or the 
anticipation of the eclipses, either to any irregularity 
occurring in the movement of the little planet or to its 

If one considers the vast size of the diameter KL, which 
according to me is some 24 thousand diameters of the 
Earth, one will acknowledge the extreme velocity of Light. 
For, supposing that KL is no more than 22 thousand of 
these diameters, it appears that being traversed in 22 
minutes this makes the speed a thousand diameters in one 
minute, that is i6j- diameters in one second or in one beat 
of the pulse, which makes more than 1 1 hundred times a 
hundred thousand toises; since the diameter of the Earth 
contains 2,865 leagues, reckoned at 25 to the degree, and 

c each 


each league is 2,282 Toises, according to the exa6t measure- 
ment which Mr. Picard made by order of the King in 1 669. 
But Sound, as I have said above, only travels 180 toises in 
the same time of one second : hence the velocity of Light 
is more than six hundred thousand times greater than that 
of Sound. This, however, is quite another thing from 
being instantaneous, since there is all the difference be- 
tween a finite thing and an infinite. Now the successive 
movement of Light being confirmed in this way, it follows, 
as I have said, that it spreads by spherical waves, like the 
movement of Sound. - 

But if the one resembles the other in this respe<5t, they 
differ in many other things; to wit, in the first production 
of the movement which causes them; in the matter in 
which the movement spreads; and in the manner in which 
it is propagated. As to that which occurs in the production 
of Sound, one knows that it is occasioned by the agitation 
undergone by an entire body, or by a considerable part of 
one, which shakes all the contiguous air. But the move- 
ment of the Light must originate as from each point of the 
luminous obje<5t, else we should not be able to perceive all 
the different parts of that obje6t, as will be more evident in 
that which follows. And I do not believe that this move- 
ment can be better explained than by supposing that all 
those of the luminous bodies which are liquid,such as flames, 
and apparently the sun and the stars, are composed of 
particles which float in a much more subtle medium which 
agitates them with great rapidity, and makes them strike 
against the particles of the ether which surrounds them, 
and which are much smaller than they. But I hold also 
that in luminous solids such as charcoal or metal made red 
hot in the fire, this same movement is caused by the violent 



agitation of the particles of the metal or of the wood ; 
those of them which are on the surface striking similarly 
against the ethereal matter. The agitation, moreover, of 
the particles which engender the light ought to be much 
more prompt and more rapid than is that of the bodies 
which cause sound, since we do not see that the tremors 
of a body which is giving out a sound are capable of giving 
rise to Light, even as the movement of the hand in the air 
is not capable of producing Sound. 

Now if one examines what this matter may be in 
which the movement coming from the luminous body is 
propagated, which I call Ethereal matter, one will see 
that it is not the same that serves for the propagation 
of Sound. For one finds that the latter is really that which 
we feel and which we breathe, and which being removed 
from any place still leaves there the other kind of matter 
that serves to convey Light. This may be proved by 
shutting up a sounding body in a glass vessel from which 
the air is withdrawn by the machine which Mr. Boyle 
has given us, and with which he has performed so many 
beautiful experiments. But in doing this of which I 
speak, care must be taken to place the sounding body on 
cotton or on feathers, in such a way that it cannot commu- 
nicate its tremors either to the glass vessel which encloses 
it, or to the machine; a precaution which has hitherto 
been negle&ed. For then after having exhausted all the 
air one hears no Sound from the metal, though it is 

One sees here not only that our air, which does not 
penetrate through glass, is the matter by which Sound 
spreads; but also that it is not the same air but another 
kind of matter in which Light spreads; since if the air is 



removed from the vessel the Light does not cease to traverse 
it as before. 

And this last point is demonstrated even more clearly by 
the celebrated experiment of Torricelli, in which the tube 
of glass from which the quicksilver has withdrawn itself, 
remaining void of air, transmits Light just the same as 
when air is in it. For this proves that a matter different 
from air exists in this tube, and that this matter must 
have penetrated the glass or the quicksilver, either one or 
the other, though they are both impenetrable to the air. 
And when, in the same experiment, one makes the vacuum 
after putting a little water above the quicksilver, one con- 
cludes equally that the said matter passes through glass or 
water, or through both. 

As regards the different modes in which I have said the 
movements of Sound and of Light are communicated, one 
may sufficiently comprehend how this occurs in the case 
of Sound if one considers that the air is of such a nature 
that it can be compressed and reduced to a much smaller 
space than that which it ordinarily occupies. And in pro- 
portion as it is compressed the more does it exert an effort 
to regain its volume; for this property along with its 
penetrability, which remains notwithstanding its com- 
pression, seems to prove that it is made up of small bodies 
which float about and which are agitated very rapidly in 
the ethereal matter composed of much smaller parts. So 
that the cause of the spreading of Sound is the effort which 
these little bodies make in collisions with one another, to 
regain freedom whe^n they are a little more squeezed to- 
gether in the circuit of these waves than elsewhere. 

But the extreme velocity of Light, and other properties 
which it has, cannot admit of such a propagation of motion, 



and I am about to show here the way in which I conceive 
it must occur. For this, it is needful to explain the pro- 
perty which hard bodies must possess to transmit move- 
ment from one to another. . 

When one takes a number of spheres of equal size, made 
of some very hard substance, and arranges them in a straight 
line, so that they touch one another, one finds, on striking 
with a similar sphere against the first of these spheres, that 
the motion passes as in an instant to the last of them, 
which separates itself from the row, without one's being 
able to perceive that the others have been stirred. And 
even that one which was used to strike remains motionless 
with them. Whence one sees that the movement passes 
with an extreme velocity which is the greater, the greater 
the hardness of the substance of the spheres. 

But it is still certain that this progression of motion is 
not instantaneous, but successive, and therefore must take 
time. For if the movement, or the disposition to move- 
ment, if you will have it so, did not pass successively 
through all these spheres, they would all acquire the 
movement at the same time, and hence would all advance 
together; which does not happen. For the last one leaves 
the whole row and acquires the speed of the one which was 
pushed. Moreover there are experiments which demon- 
strate that all the bodies which we reckon of the hardest 
kind, such as quenched steel, glass, and agate, aft as 
springs and bend somehow, not only when extended as 
rods but also when they are in the form of spheres or of 
other shapes. That is to say they yield a little in them- 
selves at the place where they are struck, and immediately 
regain their former figute. For I have found that on strik- 
ing with a ball of glass or of agate against a large and quite 



thick piece of the same substance whicn nau a flat surface, 
slightly soiled with breath or in some other way, there 
remained round marks, of smaller or larger size according 
as the blow had been weak or strong. This makes it 
evident that these substances yield where they meet, and 
spring back: and for this time must be required. 

Now in applying this kind of movement to that which 
produces Light there is nothing to hinder us from estimat- 
ing the particles of the ether to be of a substance as nearly 
approaching to perfeft hardness and possessing a springi- 
ness as prompt as we choose. It is not necessary to examine 
here the causes of this hardness, or of that springiness, 
the consideration of which would lead us too far from 
our subjeft. I will say, however, in passing that we may 
conceive that the particles of the ether, notwithstanding 
their smallness, are in turn composed of other parts and that 
their springiness consists in the very rapid movement of 
a subtle matter which penetrates them from every side 
and constrains their structure to assume such a disposition as 
to give to this fluid matter the most overt and easy pass- 
age possible. This accords with the explanation which 
Mr. Des Cartes gives for the spring, though I do not, like 
him, suppose the pores to be in the form of round hollow 
canals. And it must not be thought that in this there is 
anything absurd or impossible, it being on the contrary 
quite credible^ that it is this infinite series of different 
sizes of corpuscles, having different degrees of velocity, 
of which Nature makes use to produce so many marvellous 

But though we shall ignore the true cause of springiness 
we still see that there are many bodies which possess this 
property; and thus there is nothing strange in supposing 



that it exists also in little invisible bodies like the particles 
of the Ether. Also if one wishes to seek for any other way 
in which the movement of Light is successively com- 
municated, one will find none which agrees better, with 
uniform progression, as seems to be necessary, than the pro- 
perty of springiness; because if this movement should grow 
slower in proportion as it is shared over a greater quantity 
of matter, in moving away from the source of the light, it 
could not conserve this great velocity over great distances. 
But by supposing springiness in the ethereal matter, its 
particles will have the property of equally rapid restitution 
whether they are pushed strongly or feebly; and thus the 
propagation of Light will always go on with an equal 

And it must be known that although the particles of 
the ether are not ranged thus in straight lines, as in our 
row of spheres, but confusedly, so that one of them touches 
several others, this does not hinder them from transmitting 
their movement and from spreading it always forward. As to 
this it is to be remarked that there is a law ^-^ 

of motion serving for this propagation, and ( B ) 

verifiable by experiment. It is that when ^ ' 

a sphere, such as A here, touches several 
other similar spheres CCC, if it is struck 
by another sphere B in such a way as to 
exert an impulse against all the spheres 
CCC which touch it, it transmits to them 
the whole of its movement, and remains 
after that motionless like the sphere B. And without sup- 
posing that the ethereal particles are of spherical form (for 
I see indeed no need to suppose them so) one may well 
understand that this property of communicating an im- 


pulse does not fail to contribute to the aforesaid propaga- 
tion of movement. 

Equality of size seems * to be more necessary, because 
otherwise there ought to be some reflexion of movement 
backwards when it passes from a smaller particle to a 
larger one, according to the Laws of Percussion which I 
published some years ago. 

However, one will see hereafter that we have to suppose 
such an equality not so much as a necessity for the propaga- 
tion of light as for rendering that propagation easier and 
more powerful ; for it is not beyond the limits of prob- 
ability that the particles of the ether have been made 
equal for a purpose so important as that of light, at least 
in that vast space which is beyond the region of atmo- 
sphere and which seems to serve only to transmit the 
light of the Sun and the Stars. 

I have then shown in what manner one may conceive 
Light to spread successively, by spherical waves, and how 
it is possible that this spreading is accomplished with as 
great a velocity as that which experiments and celestial 
observations demand. Whence it may be further remarked 
that although the particles are supposed to be in con- 
tinual movement (for there are many reasons for this) the 
successive propagation of the waves cannot be hindered by 
this; because the propagation consists nowise in the trans- 
port of those particles but merely in a small agitation which 
they cannot help communicating to those surrounding, 
notwithstanding any movement which may aft on them 
causing them to be changing positions amongst themselves. 

But we must consider still more particularly the origin 
of these waves, and the manner in which they spread. 
And, first, it follows from what has been said on the pro- 



du6Hon of Light, that each little region of a luminous body, 
such as the Sun, a candle, or a burning coal, generates its 
own waves of which that region is the 
centre. Thus in the flame of a candle, 
having distinguished the points A, B, C, 
concentric circles described about each 
of these points represent the waves 
which come from them. And one must 
imagine the same about every point 
of the surface and of the part within 
the flame. 

But as the percussions at the centres 
of these waves possess no regular suc- 
cession, it must not be supposed that 
the waves themselves follow one another 
at equal distances: and if the distances marked in the figure 
appear to be such, it is rather to mark the progression of 
one and the same wave at equal intervals of time than to 
represent several of them issuing from one and the same 

After all, this prodigious quantity of waves which 
traverse one another without confusion and without effac- 
ing one another must not be deemed inconceivable; it 
being certain that one and the same particle of matter can 
serve for many waves coming from different sides or even 
from contrary dire<5tions, not only if it is struck by blows 
which follow one another closely but even for those which 
a<5t on it at the same instant. It can do so because the 
spreading of the movement is successive. This may be 
proved by the row of equal spheres of hard matter, spoken 
of above. If against this row there are pushed from two 
opposite sides at the same time two similar spheres A and 

D D, 


D,one will see each of them rebound with the same velocity 
which it had in striking, yet the whole row will remain in 


its place, although the movement has passed along its whole 
length twice over. And if these contrary movements happen 
to meet one another at the middle sphere, B, or at some 
other such as C, that sphere will yield and at as a spring 
at both sides, and so will serve at the same instant to trans- 
mit these two movements. 

But what may at first appear full strange and even in- 
credible is that the undulations produced by such small 
movements and corpuscles, should spread to such immense 
distances; as for example from the Sun or from the Stars 
to us. For the force of these waves must grow feeble in 
proportion as they move away from their origin, so that 
the aftion of each one in particular will without doubt be- 
come incapable of making itself felt to our sight. But one 
will cease to be astonished by considering how at a great 
distance from the luminous body an infinitude of waves, 
though they have issued from different points of this body, 
unite together in such a way that they sensibly compose 
one single wave only, which, consequently, ought to have 
enough force to make itself felt. Thus this infinite number 
of waves which originate at the same instant from all 
points of a fixed star, big it may be as the Sun, make 
practically only one single wave which may well have 
force enough to produce an impression on our eyes. More- 
over from each luminous point there may come many 
thousands of waves in the smallest imaginable time, by 
the frequent percussion of the corpuscles which strike the 



Ether at these points: which further contributes to render- 
ing their aftion more sensible. 

There is the further consideration in the emanation of 
these waves, that each particle of matter in which a wave 
spreads, ought not to communicate its motion only to the 
next particle which is in the straight line drawn from the 
luminous point, but that it also imparts some of it 
necessarily to all the others which touch it and which 
oppose themselves to its movement. So it arises that around 
each particle there is made 
a wave of which that 
particle is the centre. 
Thus if DCF is a wave 
emanating from the lu- 
minous point A, which 
is its centre, the particle 
B, one of those comprised 
within the sphere DCF, 
will have made its par- 
ticular or partial wave 
KCL, which will touch 
the wave DCF at C at 
the same moment that the principal wave emanating from 
the point A has arrived at DCF ; and it is clear that it 
will be only the region C of the wave KCL which will 
touch the wave DCF, to wit, that which is in the straight 
line drawn through AB. Similarly the other particles of 
the sphere DCF, such as bb> dd^ etc., will each make its 
own wave. But each of these waves can be infinitely 
feeble only as compared with the wave DCF, to the com- 
position of which all the others contribute by the part of 
their surface which is most distant from the centre A. 




One sees, in addition, that the wave DCF is determined 
by the distance attained in a certain space of time by the 
movement which started from the point A; there being no 
movement beyond this wave, though there will be in the 
space which it encloses, namely in parts of the particular 
waves, those parts which do not touch the sphere DCF. 
And all this ought not to seem fraught with too much 
minuteness or subtlety, since we shall see in the sequel 
that all the properties of Light, and everything pertain- 
ing to its reflexion and its refra6lion, can be explained 
in principle by this means. This is a matter which has 
been quite unknown to those who hitherto have begun 
to consider the waves of light, amongst whom are Mr* 
Hooke in his Micrographia^ and Father Pardies, who, 
in a treatise of which he let me see a portion, and which 
he was unable to complete as he died shortly afterward, 
had undertaken to prove by these waves the effefts of 
reflexion and refra&ion. But the chief foundation, which 
consists in the remark I have just made, was lacking in 
his demonstrations; and for the rest he had opinions very 
different from mine, as may be will appear some day if his 
writing has been preserved. 

To come to the properties of Light. We remark first 
that each portion of a wave ought to spread in such a way 
that its extremities lie always between the same straight 
lines drawn from the luminous point. Thus the portion 
BG of the wave, having the luminous point A as its centre, 
will spread into the arc CE bounded by the straight 
lines ABC, AGE. For although the particular waves pro- 
duced by the particles comprised within the space CAE 
spread also outside this space, they yet do not concur at the 
same instant to compose a wave which terminates the 



movement, as they do precisely at the circumference CE, 
which is their common tangent. 

And hence one sees the reason why light, at least if its 
rays are not reflefted or broken, spreads only by straight 
lines, so that it illuminates no object except when the path 
from its source to that obje6l is open along such lines. 
For if, for example, there were an opening BG, limited 
by opaque bodies BH, GI, the wave of light which issues 
from the point A will always be terminated by the straight 
lines AC, AE, as has just been shown ; the parts of the par- 
tial waves which spread outside the space ACE being too 
feeble to produce light there. 

Now, however small we make the opening BG, there 
is always the same reason causing the light there to pass 
between straight lines ; since this opening is always large 
enough to contain a great number of particles of the 
ethereal matter, which are of an inconceivable smallness ; 
so that it appears that each little portion of the wave 
necessarily advances following the straight line which 
comes from the luminous point. Thus then we may take 
the rays of light as if they were straight lines. 

It appears, moreover, by what has been remarked touch- 
ing the feebleness of the particular waves, that it is not 
needful that all the .particles of the Ether should be equal 
amongst themselves, though equality is more apt for the 
propagation of the movement. For it is true that inequality 
will cause a particle by pushing against another larger one 
to strive to recoil with a part of its movement; but it will 
thereby merely generate backwards towards the luminous 
point some partial waves incapable of causing light, and 
not a wave compounded of many as CE was. 

Another property of waves of light, and one of the most 



marvellous, is that when some of them come from different 
or even from opposing sides, they produce their effeft across 
one another without any hindrance. Whence also it comes 
about that a number of spectators may view different objefts 
at the same time through the same opening, and that two 
persons can at the same time see one another's eyes. Now 
according to the explanation which has been given of the 
action of light, how the waves do not destroy nor interrupt 
one another when they cross one another, these effects 
which I have just mentioned are easily conceived. But in 
my judgement they are not at all easy to explain accord- 
ing to the views of Mr. Des Cartes, who makes Light to 
consist in a continuous pressure merely tending to move- 
ment. For this pressure not being able to aft from two 
opposite sides at the same time, against bodies which 
have no inclination to approach one another, it is im- 
possible so to understand what I have been saying about 
two persons mutually seeing one another's eyes, or how 
two torches can illuminate one another. 


AVING explained the effefts of waves of 
light which spread in a homogeneous 
matter, we will examine next that which 
happens to them on encountering other 
bodies. We will first make evident how 
the Reflexion of light is explained by these 
same waves, and why it preserves equality of angles. 



Let there be a surface AB, plane and polished, of some 
metal, glass, or other body, which at first I will consider 
as perfeftly uniform (reserving to myself to deal at the end 
of this demonstration with the inequalities from which it 
cannot be exempt), and let a line AC, inclined to A&, 
represent a portion of a wave of light, the centre of which 
is so distant that this portion AC may be considered as a 
straight line; for 
I consider all this 
as in one plane, 
imagining to my- 
self that the plane 
in which this fig- 
ure is, cuts the 
sphere of the wave 
through its centre 
and intersects the 
plane AB at right 
angles. This ex- 
planation will suf- 
fice once for all. 

The piece C of 
the wave AC, will 
in a certain space of time advance as far as the plane AB 
at B, following the straight line CB, which may be sup- 
posed to come from the luminous centre, and which in 
consequence is perpendicular to AC. Now in this same 
space of time the portion A of the same wave, which has 
been hindered from communicating its movement beyond 
the plane AB, or at least partly so, ought to have con- 
tinued its movement in the matter which is above this 
plane, and this along a distance equal to CB, making its 



own partial spherical wave, according to what has been 
said above. Which wave is here represented by the cir- 
cumference SNR, the centre of which is A, and its semi- 
diameter AN equal to CB. 

If one considers further the other pieces H of the wave 
AC, it appears that they will not only have reached the 
surface AB by straight lines HK parallel to CB, but that 
in addition they will have generated in the transparent air, 
from the centres K, K, K, particular spherical waves, re- 
presented here by circumferences the semi-diameters of 
which are equal to KM, that is to say to the continuations 
of HK as far as the line, BG parallel to AC. But all these 
circumferences have as a common tangent the straight line 
BN, namely the same which is drawn from B as a tangent 
to the first of the circles, of which A is the centre, and AN 
the semi-diameter equal to BC, as is easy to see. 

It is then the line BN (comprised between B and the 
point N where the perpendicular from the point A falls) 
which is as it were formed by all these circumferences, and 
which terminates the movement which is made by the 
reflexion of the wave AC ; and it is also the place where 
the movement occurs in much greater quantity than any- 
where else. Wherefore, according to that which has been 
explained, BN is the propagation of the wave AC at the 
moment when the piece C of it has arrived at B. For there 
is no other line which like BN is a common tangent to all 
the aforesaid circles, except BG below the plane AB; 
which line BG would be the propagation of the wave if 
the movement could have spread in a medium homogene- 
ous with that which is above the plane. And if one wishes 
to see how the wave AC has come successively to BN, one 
has only to draw in the same figure the straight lines KO 



parallel to BN, and the straight lines KL parallel to AC. 
Thus one will see that the straight wave AC has become 
broken up into all the OKL parts successively, and that it 
has become straight again at NB. 

Now it is apparent here that the angle of reflexion is 
made equal to the angle of incidence. For the triangles 
ACB, BNA being re6tangular and having the side AB 
common, and the side CB equal to NA, it follows that the 
angles opposite to these sides will be equal, and therefore 
also the angles CBA, NAB. But as CB, perpendicular to 
CA, marks the direction of the incident ray, so AN, per- 
pendicular to the wave BN, marks the direftion of the 
reflected ray ; hence these rays are equally inclined to the 
plane AB. 

But in considering the preceding demonstration, one 
might aver that it is indeed true that BN is the common 
tangent of the circular waves in the plane of this figure, 
but that these waves, being in truth spherical, have still an 
infinitude of similar tangents, namely all the straight lines 
which are drawn from the point B in the surface generated 
by the straight line BN about the axis BA. It remains, 
therefore, to demonstrate that there is no difficulty herein : 
and by the same argument one will see why the incident 
ray and the reflected ray are always in one and the same 
plane perpendicular to the reflecting plane. I say then that 
the wave AC, being regarded only as a line, produces no 
light. For a visible ray of light, however narrow it may be, 
has always some width, and consequently it is necessary, 
in representing the wave whose progression constitutes the 
ray, to put instead of a line AC some plane figure such as 
the circle HC in the following figure, by supposing, as we 
have done, the luminous point to be infinitely distant. 

E Now 


Now it is easy to see, following the preceding demonstra- 
tion, that each small piece of this wave HC having arrived 
at the plane AB, and there generating each one its parti- 
cular wave, these will all have, when C arrives at B, a com- 
mon plane which will touch them, namely a circle BN 
similar to CH; and this will be intersected at its middle 
and at right angles by the same plane which likewise in- 
tersedts the circle CH and the ellipse AB. 

One sees also that the said spheres of the partial waves 
cannot have any common tangent plane other than the 

circle BN; so that it will be this plane where there will 
be more reflected movement than anywhere else, and 
which will therefore carry on the light in continuance from 
the wave CH. 

I have also stated in the preceding demonstration that the 
movement of the piece A of the incident wave is not able 
to communicate itself beyond the plane AB, or at least not 
wholly. Whence it is to be remarked that though the 
movement of the ethereal matter might communicate itself 
partly to that of the reflecting body, this could in nothing 
alter the velocity of progression of the waves, on which 



the angle of reflexion depends. For a slight percussion 
ought to generate waves as rapid as strong percussion in 
the same matter. This comes about from the property of 
bodies which aft as springs, of which we have spoken above; 
namely that whether compressed little or much they recoil 
in equal times. Equally so in every reflexion of the light, 
against whatever body it may be, the angles of reflexion 
and incidence ought to be equal notwithstanding that the 
body might be of such a nature that it takes away a portion 
of the movement made by the incident light. And experi- 
ment shows that in fa6t there is no polished body the re- 
flexion of which does not follow this rule. 

But the thing to be above all remarked in our demon- 
stration is that it does not require that the reflecting surface 
should be considered as a uniform plane, as has been sup- 
posed by all those who have tried to explain the effects of 
reflexion ; but only an evenness such as may be attained by 
the particles of the matter of the reflecting body being set 
near to one another; which particles are larger than those of 
the ethereal matter, as will appear by what we shall say in 
treating of the transparency and opacity of bodies. For 
the surface consisting thus of particles put together, and 
the ethereal particles being above, and smaller, it is evident 
that one could not demonstrate the equality of the angles 
of incidence and reflexion by similitude to that which 
happens to a ball thrown against a wall, of which writers 
have always made use. In our way, on the other hand, 
the thing is explained without difficulty. For the smallness 
of the particles of quicksilver, for example, being such that 
one must conceive millions of them, in the smallest visible 
surface proposed, arranged like a heap of grains of sand 
which has been flattened as much as it is capable of being, 



this surface then becomes for our purpose as even as a 
polished glass is : and, although it always remains rough 
with respeft to the particles of the Ether it is evident that 
the centres of all the particular spheres of reflexion, of which 
we have spoken, are almost in one uniform plane, and that 
thus the common tangent can fit to them as perfectly as is 
requisite for the production of light. And this alone is re- 
quisite, in our method of demonstration, to cause equality 
of the said angles without the remainder of the movement 
refle&ed from all parts being able to produce any contrary 


N the same way as the effects of Reflexion 
have been explained by waves of light re- 
fledled at the surface of polished bodies, we 
will explain transparency and thephenomena 
of refraction by waves which spread within 
and across diaphanous bodies, both solids, 
such as glass, and liquids, such as water, oils, etc. But in 
order that it may not seem strange to suppose this passage 
of waves in the interior of these bodies, I will first 
show that one may conceive it possible in more than 
one mode. 

First, then, if the ethereal matter cannot penetrate 
transparent bodies at all, their own particles would be able 
to communicate successively the movement of the waves, 
the same as do those of the Ether, supposing that, like 
those, they are of a nature to aft as a spring. And this is 



easy to conceive as regards water and other transparent 
liquids, they being composed of detached particles. But 
it may seem more difficult as regards glass and other trans- 
parent and hard bodies, because their solidity does not seem 
to permit them to receive movement except in their whole 
mass at the same time. This, however, is not necessary 
because this solidity is not such as it appears to us, it being 
probable rather that these bodies are composed of particles 
merely placed close to one another and held together by 
some pressure from without of some other matter, and by 
the irregularity of their shapes. For primarily their rarity 
is shown by the facility with which there passes through 
them the matter of the vortices of the magnet, and that 
which causes gravity. Further, one cannot say that these 
bodies are of a texture similar to that of a sponge or of light 
bread, because the heat of the fire makes them flow and 
thereby changes the situation of the particles amongst 
themselves. It remains then that they are, as has been said, 
assemblages of particles which touch one another without 
constituting a continuous solid. This being so, the move- 
ment which these particles receive to carry on the waves of 
light, being merely communicated from some of them to 
others, without their going for that purpose out of their 
places or without derangement, it may very well produce 
its effe6l without prejudicing in any way the apparent 
solidity of the compound. 

By pressure from without, of which I have spoken, 
must not be understood that of the air, which would not 
be sufficient, but that of some other more subtle matter, a 
pressure which I chanced upon by experiment long ago, 
namely in the case of water freed from air, which remains 
suspended in a tube open at its lower end, notwithstanding 



that the air has been removed from the vessel in which 
this tube is enclosed. 

One can then in this way conceive of transparency in a 
solid without any necessity that the ethereal matter which 
serves for light should pass through it, or that it should 
find pores in which to insinuate itself. But the truth is 
that this matter not only passes through solids, but does so 
even with great facility ; of which the experiment of 
Torricelli, above cited, is already a proof. Because on 
the quicksilver and the water quitting the upper part of 
the glass tube, it appears that it is immediately filled with 
ethereal matter, since light passes across it. But here is 
another argument which proves this ready penetrability, 
not only in transparent bodies but also in all others. 

When light passes across a hollow sphere of glass, closed 
on all sides, it is certain that it is full of ethereal matter, 
as much as the spaces outside the sphere. And this ethereal 
matter, as has been shown above, consists of particles which 
just touch one another. If then it were enclosed in the 
sphere in such a way that it could not get out through 
the pores of the glass, it would be obliged to follow the 
movement of the sphere when one changes its place : and 
it would require consequently almost the same force to 
impress a certain velocity on this sphere, when placed on 
a horizontal plane, as if it were full of water or perhaps 
of quicksilver : because every body resists the velocity of 
the motion which one would give to it, in proportion to 
the quantity of matter which it contains, and which is 
obliged to follow this motion. But on the contrary one 
finds that the sphere resists the impress of movement only in 
proportion to the quantity of matter of the glass of which 
it is made. Then it must be that the ethereal matter which 



is inside is not shut up, but flows through it with very 
great freedom. We shall demonstrate hereafter that by 
this process the same penetrability may be inferred also as 
relating to opaque bodies. 

The second mode then of explaining transparency, and 
one which appears more probably true, is by saying that the 
waves of light are carried on in the ethereal matter, which 
continuously occupies the interstices or pores of transpar- 
ent bodies. For since it passes through them continuously 
and freely, it follows that they are always full of it. And 
one may even show that these interstices occupy much 
more space than the coherent particles which constitute 
the bodies. For if what we have just said is true : that force 
is required to impress a certain horizontal velocity on bodies 
in proportion as they contain coherent matter; and if the 
proportion of this force follows the law of weights, as is 
confirmed by experiment, then the quantity of the con- 
stituent matter of bodies also follows the proportion of 
their weights. Now we see that water weighs only one 
fourteenth part as much as an equal portion of quicksilver : 
therefore the matter of the water does not occupy the 
fourteenth part of the space which its mass obtains. It 
must even occupy much less of it, since quicksilver is less 
heavy than gold, and the matter of gold is by no means 
dense, as follows from the fa6l that the matter of the 
vortices of the magnet and of that which is the cause of 
gravity pass very freely through it. 

But it may be objedled here that if water is a body of 
so great rarity, and if its particles occupy so small a 
portion of the space of its apparent bulk, it is very strange 
how it yet resists Compression so strongly without per- 
mitting itself to be condensed by any force which one has 



hitherto essayed to employ, preserving even its entire 
liquidity while subjeded to this pressure. 

This is no small difficulty. It may, however, be resolved 
by saying that the very violent and rapid motion of the 
subtle matter which renders water liquid, by agitating the 
particles of which it is composed, maintains this liquidity 
in spite of the pressure which hitherto any one has been 
minded to apply to it. 

The rarity of transparent bodies being then such as we 
have said, one easily conceives that the waves might be 
carried on in the ethereal matter which fills the inter- 
stices of the particles. And, moreover, one may believe 
that the progression of these waves ought to be a little 
slower in the interior of bodies, by reason of the small 
detours which the same particles cause. In which differ- 
ent velocity of light I shall show the cause of refraftion 
to consist. 

Before doing so, I will indicate the third and last mode 
in which transparency may be conceived; which is by 
supposing that the movement of the waves of light is 
transmitted indifferently both in the particles of the 
ethereal matter which occupy the interstices of bodies, 
and in the particles which compose them, so that the 
movement passes from one to the other. And it will be 
seen hereafter that this hypothesis serves excellently to 
explain the double refra6tion of certain transparent bodies. 

Should it be objected that if the particles of the ether 
are smaller than those of transparent bodies (since they 
pass through their intervals), it would follow that they 
can communicate to them but little of their movement, it 
may be replied that the particles of these bodies are in 
turn composed of still smaller particles, and so it will be 



these secondary particles which will receive the movement 
from those of the ether. 

Furthermore, if the particles of transparent bodies have 
a recoil a little less prompt than that of the ethereal 
particles, which nothing hinders us from supposing, it 
will again follow that the progression of the waves of light 
will be slower in the interior of such bodies than it is 
outside in the ethereal matter. 

All this I have found as most probable for the mode in 
which the waves of light pass across transparent bodies. 
To which it must further be added in what respe<5l these 
bodies differ from those which are opaque; and the more 
so since it might seem because of the easy penetration of 
bodies by the ethereal matter, of which mention has been 
made, that there would not be any body that was not trans- 
parent. For by the same reasoning about the hollow sphere 
which I have employed to prove the smallness of the density 
of glass and its easy penetrability by the ethereal matter, 
one might also prove that the same penetrability obtains 
for metals and for every other sort of body. For this 
sphere being for example of silver, it is certain that it 
contains some of the ethereal matter which serves for light, 
since this was there as well as in the air when the opening 
of the sphere was closed. Yet, being closed and placed 
upon a horizontal plane, it resists the movement which one 
wishes to give to it, merely according to the quantity of 
silver of which it is made; so that one must conclude, as 
above, that the ethereal matter which is enclosed does not 
follow the movement of the sphere; and that therefore sil- 
ver, as well as glass, is very easily penetrated by this matter. 
Some of it is therefore present continuously and in quan- 
tities between the particles of silver and of all other opaque 

F bodies: 


bodies: and since it serves for the propagation of light it 
would seem that these bodies ought also to be transparent, 
which however is not the case. 

Whence then, one will say, does their opacity come? 
Is it because the particles which compose them are soft; 
that is to say, these particles being composed of others 
that are smaller, are they capable of changing their figure 
on receiving the pressure of the ethereal particles, the mo- 
tion of which they thereby damp, and so hinder the con- 
tinuance of the waves of light? That cannot be: for if the 
particles of the metals are soft, how is it that polished 
silver and mercury refleft light so strongly? What I find 
to be most probable herein, is to say that metallic bodies, 
which are almost the only really opaque ones, have mixed 
amongst their hard particles some soft ones; so that some 
serve to cause reflexion and the others to hinder trans- 
parency; while, on the other hand, transparent bodies con- 
tain only hard particles which have the faculty of recoil, 
and serve together with those of the ethereal matter for 

the propagation of the waves of 
light, as has been said. 

Let us pass now to the ex- 
planation of the effects of Re- 
fraction, assuming, as we have 
done, the passage of waves of 
light through transparent 
bodies, and the diminution of 
velocity which these same waves 
suffer in them. 
The chief property of Refraction is that a ray of light, 
such as AB, being in the air, and falling obliquely upon 
the polished surface of a transparent body, such as FG, is 



broken at the point of incidence B, in such a way that 
with the straight line DBE which cuts the surface per- 
pendicularly it makes an angle CBE less than ABD which 
it made with the same perpendicular when in the air. And 
the measure of these angles is found by describing, about 
the point B, a circle which cuts the radii AB, BC. For 
the perpendiculars AD, CE, let fall from the points of 
intersection upon the straight line DE, which are called 
the Sines of the angles ABD, CBE, have a certain ratio 
between themselves; which ratio is always the same for all 
inclinations of the incident ray, at least for a given trans- 
parent body. This ratio is, in glass, very nearly as 3 to 2; 
and in water very nearly as 4 to 3; and is likewise different 
in other diaphanous bodies. 

Another property, similar to this, is that the refra6lions 
are reciprocal between the rays entering into a transparent 
body and those which are leaving it. That is to say that 
if the ray AB 
in entering the 
t r ansp arent 
body is refradted 
into BC, then 
likewise CB 
being taken as 
a ray in the 
interior of this 
body will be 
refrafted, on 
passing out, in- 
to BA. 

To explain then the reasons of these phenomena accord- 
ing to our principles, let AB be the straight line which 



represents a plane surface bounding the transparent sub- 
stances which lie towards C and towards N. When I say 
plane, that does not signify a perfeft evenness, but such 
as has been understood in treating of reflexion, and for the 
same reason. Let the line AC represent a portion of a 
wave of light, the centre of which is supposed so distant 
that this portion may be considered as a straight line. 
The piece C, then, of the wave AC, will in a certain space 
of time have advanced as far as the plane AB following 
the straight line CB, which may be imagined as coming 
from the luminous centre, and which consequently will 
cut AC at right angles. Now in the same time the piece 
A would have come to G along the straight line AG, 
equal and parallel to CB; and all the portion of wave AC 
would be at GB if the matter of the transparent body 
transmitted the movement of the wave as quickly as the 
matter of the Ether. But let us suppose that it transmits 
this movement less quickly, by one-third, for instance. 
Movement will then be spread from the point A, in the 
matter of the transparent body through a distance equal 
to two-thirds of CB, making its own particular spherical 
wave according to what has been said before. This wave 
is then represented by the circumference SNR, the centre 
of which is A, and its semi-diameter equal to two-thirds of 
CB. Then if one considers in order the other pieces H of 
the wave AC, it appears that in the same time that the piece 
C reaches B they will not only have arrived at the surface 
AB along the straight lines HK parallel to CB, but that, 
in addition, they will have generated in the diaphanous 
substance from the centres K, partial waves, represented 
here by circumferences the semi-diameters of which are 
equal to two-thirds of the lines KM, that is to say, to 



two-thirds of the prolongations of HK down to the straight 
line BG; for these semi-diameters would have been equal 
to entire lengths of KM if the two transparent substances 
had been of the same penetrability. 

Now all these circumferences have for a common tangent 
the straight line BN; namely the same line which is drawn 
as a tangent from the point B to the circumference SNR 
which we considered first. For it is easy to see that all 
the other circumferences will touch the same BN, from 
B up to the point of contact N, which is the same point 
where AN falls perpendicularly on BN. 

It is then BN, which is formed by small arcs of these cir- 
cumferences, which terminates the movement that the wave 
AC has communicated within the transparent body, and 
where this movement occurs in much greater amount than 
anywhere else. And for that reason this line, in accordance 
with what has been said more than once, is the propaga- 
tion of the wave AC at the moment when its piece C has 
reached B. For there is no other line below the plane 
AB which is, like BN, a common tangent to all these 
partial waves. And if one would know how the wave AC 
has come progressively to BN, it is necessary only to draw 
in the same figure the straight lines KO parallel to BN, 
and all the lines KL parallel to AC. Thus one will see 
that the wave CA, from being a straight line, has become 
broken in all the positions LKO successively, and that it 
has again become a straight line at BN. This being 
evident by what has already been demonstrated, there is 
no need to explain it further. 

Now, in the same figure, if one draws EAF, which cuts 
the plane AB at right angles at the point A, since AD is 
perpendicular to the wave AC, it will be DA which will 



mark the ray of incident light, and AN which was per- 
pendicular to BN, the refra6led ray: since the rays are 
nothing else than the straight lines along which the 
portions of the waves advance. 

Whence it is easy to recognize this chief property of re- 
fradtion, namely that the Sine of the angle DAE has always 
the same ratio to the Sine of the angle NAF, whatever be 
the inclination of the ray DA: and that this ratio is the 
same as that of the velocity of the waves in the transparent 
substance which is towards AE to their velocity in the 
transparent substance towards AF. For, considering AB 
as the radius of a circle, the Sine of the angle BAG is BC, 
and the Sine of the angle ABN is AN. But the angle 
BAG is equal to DAE, since each of them added to CAE 
makes a right angle. And the angle ABN is equal to 
NAF, since each of them with BAN makes a right angle. 
Then also the Sine of the angle DAE is to the Sine of 
NAF as BC is to AN. But the ratio of BC to AN was the 
same as that of the velocities of light in the substance 
which is towards AE and in that which is towards AF; 
therefore also the Sine of the angle DAE will be to the 
Sine of the angle NAF the same as the said velocities 
of light. 

To see, consequently, what the refradtion will be when 
the waves of light pass into a substance in which the 
movement travels more quickly than in that from which 
they emerge (let us again assume the ratio of 3 to 2), it 
is only necessary to repeat all the same construction and 
demonstration which we have just used, merely substitut- 
ing everywhere f instead off. And it will be found by the 
same reasoning, in this other figure, that when the piece 
C of the wave AC shall have reached the surface AB at B, 



all the portions of the wave AC will have advanced as far 
as BN, so that BC the perpendicular on AC is to AN the 
perpendicular on BN as 2 to 3. And there will finally be 
this same ratio of 2 to 3 between the Sine of the angle 
BAD and the Sine of the angle FAN. 

Hence one sees the reciprocal relation of the refra6tions 
of the ray on entering and on leaving one and the same 
transparent body: namely that if NA falling on the ex- 
ternal surface AB 
is refradted into the 
direction AD, so 
the ray AD will be 
refrafted on leaving 
the transparent 
body into the direc- 
tion AN. 

One sees also 
the reason for a 
noteworthy acci- 
dent which hap- 
pens in this re- 
fraftion : which is 
this, that after a certain obliquity of the incident ray DA, 
it begins to be quite unable to penetrate into the other 
transparent substance. For if the angle DAQj>r CBA is 
such that in the triangle ACB, CB is equal to f of AB, 
or is greater, then AN cannot form one side of the triangle 
ANB, since it becomes equal to or greater than AB: so 
that the portion of wave BN cannot be found anywhere, 
neither consequently can AN, which ought to be per- 
pendicular to it. And thus the incident ray DA does not 
then pierce the surface AB. 



When the ratio of the velocities of the waves is as two 
to three, as in our example, which is that which obtains 
for glass and air, the angle DAQ^jrnust be more than 
48 degrees 1 1 minutes in order that the ray DA may be 
able to pass by refra<5lion. And when the ratio of the 
velocities is as 3 to 4, as it is very nearly in water and 
air, this angle DAQjnust exceed 41 degrees 24 minutes. 
And this accords perfe6tly with experiment. 

But it might here be asked : since the meeting of the 
wave AC against the surface AB ought to produce move- 
ment in the matter which is on the other side, why does 
no light pass there ? To which the reply is easy if one 
remembers what has been said before. For although it 
generates an infinitude of partial waves in the matter which 
is at the other side of AB, these waves never have a 
common tangent line (either straight or curved) at the 
same moment; and so there is no line terminating the 
propagation of the wave AC beyond the plane AB, nor 
any place where the movement is gathered together in 
sufficiently great quantity to produce light. And one will 
easily see the truth of this, namely that CB being larger 
than J- of AB, the waves excited beyond the plane AB 
will have no common tangent if about the centres K one 
then draws circles having radii equal to -f- of the lengths 
LB to which they correspond. For all these circles will 
be enclosed in one another and will all pass beyond the 
point B. 

Now it is to be remarked that from the moment when 
the angle DAQJs smaller than is requisite to permit the 
refra&ed ray DA to pass into the other transparent sub- 
stance, one finds that the interior reflexion which occurs 
at the surface AB is much augmented in brightness, as 



is easy to realize by experiment with a triangular prism; 
and for this our theory can afford this reason. When 
the angle DAQ_is still large enough to enable the ray 
DA to pass, it is evident that the light from the portion 
AC of the wave is colle6ted in a minimum space when 
it reaches BN. It appears also that the wave BN be- 
comes so much the smaller as the angle CBA or DAQ 
is made less, until when the latter is diminished to 
the limit indicated a little previously, this wave BN is 
collected together always at one point. That is to say, 
that when the piece C of the wave AC has then arrived 
at B, the wave BN which is the propagation of AC is 
entirely reduced to the same point B. Similarly when 
the piece H has reached K, the part AH is entirely reduced 
to the same point K. This makes it evident that in pro- 
portion as the wave CA comes to meet the surface AB, 
there occurs a great quantity of movement along that 
surface; which movement ought also to spread within 
the transparent body and ought to have much re-enforced 
the partial waves which produce the interior reflexion 
against the surface AB, according to the laws of reflexion 
previously explained. 

And because a slight diminution of the angle of incid- 
ence DAQ^causes the wave BN, however great it was, 
to be reduced to zero, (for this angle being 49 degrees 1 1 
minutes in the glass, the angle BAN is still 1 1 degrees 21 
minutes, and the same angle being reduced by one degree 
only the angle BAN is reduced to zero, and so the wave 
BN reduced to a point) thence it comes about that the 
interior reflexion from being obscure becomes suddenly 
bright, so soon as the angle of incidence is such that it no 
longer gives passage to the refra6tion. 

G Now 


Now as concerns ordinary external reflexion, that is to 
say which occurs when the angle of incidence DAQjs 
still large enough to enable the refrafted ray to penetrate 
beyond the surface AB, this reflexion should occur against 
the particles of the substance which touches the trans- 
parent body on its outside. And it apparently occurs against 
the particles of the air or others mingled with the ethereal 
particles and larger than they. So on the other hand the 
external reflexion of these bodies occurs against the par- 
ticles which compose them, and which are also larger 
than those of the ethereal matter, since the latter flows 
in their interstices. It is true that there remains here 
some difficulty in those experiments in which this interior 
reflexion occurs without the particles of air being able to 
contribute to it, as in vessels or tubes from which the air 
has been extracted. 

Experience, moreover, teaches us that these two re- 
flexions are of nearly equal force, and that in different 
transparent bodies they are so much the stronger as the 
refra&ion of these bodies is the greater. Thus one sees 
manifestly that the reflexion of glass is stronger than that 
of water, and that of diamond stronger than that of glass. 

I will finish this theory of refraction by demonstrating 
a remarkable proposition which depends on it; namely, 
that a ray of light in order to go from one point to 
another, when these points are in different media, is re- 
fradled in such wise at the plane surface which joins these 
two media that it employs the least possible time: and 
exactly the same happens in the case of reflexion against a 
plane surface. Mr. Fermat was the first to propound this 
property of refraction, holding with us, and directly 
counter to the opinion of Mr, Des Cartes, that light passes 



more slowly through glass and water than through air. 
But he assumed besides this a constant ratio of Sines, 
which we have just proved by these different degrees of 
velocity alone: or rather, what is equivalent, he assumed 
not only that the velocities were different but that the light 
took the least time possible for its passage, and thence 
deduced the constant ratio of the Sines. His demonstra- 
tion, which may be seen in his printed works, and in the 
volume of letters of Mr. Des Cartes, is very long ; where- 
fore I give here another 
which is simpler and 

Let KF be the plane 
surface; A the point in 
the medium which the - 
light traverses more 
easily, as the air ; C the 
point in the other which 
is more difficult to pene- 
trate, as water. And sup- 
pose that a ray has come 
from A, by B, to C, hav- 
ing been refraded at B according to the law demon- 
strated a little before; that is to say that, having drawn 
PBQ, which cuts the plane at right angles, let the sine 
of the angle ABP have to the sine of the angle CBQ 
the same ratio as the velocity of light in the medium 
where A is to the velocity of light in the medium 
where C is. It is to be shown that the time of passage 
of light along AB and BC taken together, is the shortest 
that can be. Let us assume that it may have come by 
other lines, and, in the first place, along AF, FC, so 



that the point of refradtion F may be further from B 
than the point A; and let AO be a line perpendicular 
to AB, and FO parallel to AB ; BH perpendicular to 
FO, and FG to BC. 

Since then the angle HBF is equal to PBA, and the 
angle BFG equal to QBC, it follows that the sine of the 
angle HBF will also have the same ratio to the sine of 
BFG, as the velocity of light in the medium A is to its 
velocity in the medium C. But these sines are the straight 
lines HF, BG, if we take BF as the semi-diameter of a 
circle. Then these lines HF, BG, will bear to one another 
the said ratio of the velocities. And, therefore, the time of 
the light along HF, supposing that the ray had been OF, 
would be equal to the time along BG in the interior of 
the medium C. But the time along AB is equal to the 
time along OH ; therefore the time along OF is equal to 
the time along AB, BG. Again the time along FC is 
greater than that along GC ; then the time along OFC will 
be longer than that along ABC. But AF is longer than 
OF, then the time along AFC will by just so much 
more exceed the time along ABC. 

Now let us assume that the ray has come from A to C 
along AK, KC ; the point of refraction K being nearer 
to A than the point B is; and let CN be the perpendi- 
cular upon BC, KN parallel to BC : BM perpendicular 
upon KN, and KL upon BA. 

Here BL and KM are the sines of angles BKL, KBM; 
that is to say, of the angles PBA, QBC ; and therefore 
they are to one another as the velocity of light in the 
medium A is to the velocity in the medium C. Then 
the time along LB is equal to the time along KM ; and 
since the time along BC is equal to the time along MN, the 



time along LBC will be equal to the time along KMN. 
But the time along AK is longer than that along AL : 
hence the time along AKN is longer than that along 
ABC. And KC being longer than KN, the time along 
AKC will exceed, by as much more, the time along ABC. 
Hence it appears that the time along ABC is the shortest 
possible; which was to be proven. 



E have shown how the movement 
which constitutes light spreads by 
spherical waves in any homogeneous 
matter. And it is evident that when 
the matter is not homogeneous, but 
of such a constitution that the move- 
ment is communicated in it more rapidly toward one side 
than toward another, these waves cannot be spherical : but 
that they must acquire their figure according to the differ- 
ent distances over which the successive movement passes 
in equal times. 

It is thus that we shall in the first place explain the 
refractions which occur in the air, which extends from 
here to the clouds and beyond. The effefts of which re- 
fradions are very remarkable; for by them we often see 
objects which the rotundity of the Earth ought otherwise 
to hide; such as Islands, and the tops of mountains when 
one is at sea. Because also of them the Sun and the Moon 
appear as risen before in fa<ft they have, and appear to set 



later: so that at times the Moon has been seen eclipsed 
while the Sun appeared still above the horizon. And so also 
the heights of the Sun and of the Moon, and those of all the 
Stars always appear a little greater than they are in reality, 
because of these same refrations, as Astronomers know. 
But there is one experiment which renders this refraftion 
very evident; which is that of fixing a telescope on some 
spot so that it views an objet, such as a steeple or a 
house, at a distance of half a league or more. If then you 
look through it at different hours of the day, leaving it 
always fixed in the same way, you will see that the same 
spots of the object will not always appear at the middle 
of the aperture of the telescope, but that generally in the 
morning and in the evening, when there are more vapours 
near the Earth, these objefts seem to rise higher, so that 
the half or more of them will no longer be visible ; and 
so that they seem lower toward mid-day when these 
vapours are dissipated. 

Those who consider refraction to occur only in the 
surfaces which separate transparent bodies of different 
nature, would find it difficult to give a reason for all that 
I have just related; but according to our Theory the thing 
is quite easy. It is known that the air which surrounds 
us, besides the particles which are proper to it and which 
float in the ethereal matter as has been explained, is full 
also of particles of water which are raised by the acftion of 
heat; and it has been ascertained further by some very 
definite experiments that as one mounts up higher the 
density of air diminishes in proportion. Now whether the 
particles of water and those of air take part, by means of 
the particles of ethereal matter, in the movement which 
constitutes light, but have a less prompt recoil than these, 



or whether the encounter and hindrance which these par- 
ticles of air and water offer to the propagation of movement 
of the ethereal progress, retard the progression, it follows 
that both kinds of particles flying amidst the ethereal 
particles, must render the air, from a great height down 
to the Earth, gradually less easy for the spreading of the 
waves of light. 

Whence the configuration of the waves ought to be- 

come nearly such as this figure represents: namely, if A is 
a light, or the visible point of a steeple, the waves which 
start from it ought to spread more widely upwards and 
less widely downwards, but in other directions more or less 
as they approximate to these two extremes. This being 
so, it necessarily follows that every line intersecting one 
of these waves at right angles will pass above the point A, 
always excepting the one line which is perpendicular to 
the horizon. 



Let BC be the wave which brings the light to the 
spe6lator who is at B, and let BD be the straight line 
which intersects this wave at right angles. Now because 
the ray or straight line by which we judge the spot where 
the object appears to us is nothing else than the perpendic- 
ular to the wave that reaches our eye, as will be under- 
stood by what was said above, it is manifest that the point 

A will be per- 
c e i ved as 
being in the 
line BD, and 
higher than 
in faft it is. 

if the Earth 
be AB, and 
the top of the 
CD, which 
probably is 
not a well de- 
fined spheri- 
cal surface 

(since we know that the air becomes rare in proportion 
as one ascends, for above there is so much less of it to 
press down upon it), the waves of light from the sun 
coming, for instance, in such a way that so long as they 
have not reached the Atmosphere CD the straight line AE 
interse<5ts them perpendicularly, they ought, when they 
enter the Atmosphere, to advance more quickly in elevated 
regions than in regions nearer to the Earth. So that if 



CA is the wave which brings the light to the speftator at 
A, its region C will be the furthest advanced ; and the 
straight line AF, which intersects this wave at right angles, 
and which determines the apparent place of the Sun, will 
pass above the real Sun, which will be seen along the line 
AE. And so it may occur that when it ought not to be 
visible in the absence of vapours, because the line AE 
encounters the rotundity of the Earth, it will be perceived 
in the line AF by refra6lion. But this angle EAF is 
scarcely ever more than half a degree because the attenua- 
tion of the vapours alters the waves of light but little. 
Furthermore these refraftions are not altogether constant 
in all weathers, particularly at small elevations of 2 or 3 
degrees; which results from the different quantity of 
aqueous vapours rising above the Earth. 

And this same thing is the cause why at certain times 
a distant obje6t will be hidden behind another less distant 
one, and yet may at another time be able to be seen, 
although the spot from which it is viewed is always the 
same. But the reason for this effeft will be still more 
evident from what we are going to remark touching the 
curvature of rays. It appears from the things explained 
above that the progression or propagation of a small part 
of a wave of light is properly what one calls a ray. Now 
these rays, instead of being straight as they are in homo- 
geneous media, ought to be curved in an atmosphere of 
unequal penetrability. For they necessarily follow from 
the objeft to the eye the line which intersefts at right 
angles all the progressions of the waves, as in the first 
figure the line AEB does, as will be shown hereafter ; and 
it is this line which determines what interposed bodies 
would or would not hinder us from seeing the objed. For 

H although 


although the point of the steeple A appears raised to D, it 
would yet not appear to the eye B if the tower H was 
between the two, because it crosses the curve AEB. But 
the tower E, which is beneath this curve, does not hinder 
the point A from being seen. Now according as the air 
near the Earth exceeds in density that which is higher, 
the curvature of the ray AEB becomes greater: so that 
at certain times it passes above the summit E, which 

allows the point A to 
be perceived by the 
eye at B ; and at other 
times it is intercepted 
: by the same tower E 
which hides A from 
this same eye. 
: But to demonstrate 
this curvature of the 
rays conformably to all 
our preceding Theory, 
let us imagine that AB 
is a small portion of a 
wave of light coming 
from the side C, which 
we may consider as a straight line. Let us also suppose 
that it is perpendicular to the Horizon, the portion B 
being nearer to the Earth than the portion A; and that 
because the vapours are less hindering at A than at B, 
the particular wave which comes from the point A spreads 
through a certain space AD while the particular wave 
which starts from the point B spreads through a shorter 
space BE; AD and BE being parallel to the Horizon. 
Further, supposing the straight lines FG, HI, etc., to be 



drawn from an infinitude of points in the straight line AB 
and to terminate on the line DE (which is straight 
or may be considered as such), let the different penetra- 
bilities at the different heights in the air between A and 
B be represented by all these lines ; so that the particular 
wave, originating from the point F, will spread across the 
space FG, and that from the point H across the space HI, 
while that from the point A spreads across the space AD. 
Now if about the centres A, B, one describes the circles 
DK, EL, which represent the spreading of the waves which 
originate from these two points, and if one draws the 
straight line KL which touches these two circles, it is 
easy to see that this same line will be the common tangent 
to all the other circles drawn about the centres F, H, etc. ; 
and that all the points of contact will fall within that part 
of this line which is comprised between the perpendiculars 
AK, BL. Then it will be the line KL which will terminate 
the movement of the particular waves originating from 
the points of the wave AB ; and this movement will be 
stronger between the points KL, than anywhere else at the 
same instant, since an infinitude of circumferences concur 
to form this straight line; and consequently KL will be 
the propagation of the portion of wave AB, as has been 
said in explaining reflexion and ordinary refraction. Now 
it appears that AK and BL dip down toward the side 
where the air is less easy to penetrate: for AK being 
longer than BL, and parallel to it, it follows that the lines 
AB and KL, being prolonged, would meet at the side L. 
But the angle K is a right angle: hence KAB is necessarily 
acute, and consequently less than DAB. If one investigates 
in the same way the progression of the portion of the 
wave KL, one will find that after a further time it has 



arrived at MN in such a manner that the perpendiculars 
KM, LN, dip down even more than do AK, BL. And 
this suffices to show that the ray will continue along the 
curved line which intersefts all the waves at right angles, 
as has been said. 



IJHERE is brought from Iceland, which is 
an Island in the North Sea, in the latitude 
of 66 degrees, a kind of Crystal or trans- 
parent stone, very remarkable for its figure 
and other qualities, but above all for its 
strange refractions. The causes of this 
have seemed to me to be worthy of being carefully in- 
vestigated, the more so because amongst transparent bodies 
this one alone does not follow the ordinary rules with re- 
sped: to rays of light. I have even been under some neces- 
sity to make this research, because the refractions of this 
Crystal seemed to overturn our preceding explanation of 
regular refration; which explanation, on the contrary, they 
strongly confirm, as will be seen after they have been 
brought under the same principle. In Iceland are found 
great lumps of this Crystal, some of which I have seen of 4 
or 5 pounds. But it occurs also in other countries, for I have 
had some of the same sort which had been found in France 
near the town of Troyes in Champagne, and some others 
which came from the Island of Corsica, though both were 



less clear and only in little bits, scarcely capable of letting 
any effeCt of refraction be observed. 

2. The first knowledge which the public has had about 
it is due to Mr. Erasmus Bartholinus, who has given a 
description of Iceland Crystal and of its chief phenomena. 
But here I shall not desist from giving my own, both for 
the instruction of those who may not have seen his book, 
and because as respects some of these phenomena there is 
a slight difference between his observations and those 
which I have made : for I have applied myself with great 
exactitude to examine these properties of refraction, in 
order to be quite sure before undertaking to explain the 
causes of them. 

3. As regards the hardness of this stone, and the 
property which it has of being easily split, it must be 
considered rather as a species of Talc than of Crystal. 
For an iron spike effeCts an entrance into it as easily as 
into any other Talc or Alabaster, to which it is equal in 

4. The pieces of it which are found have the figure of an 
oblique parallelepiped ; each of the six faces being a parallel- 
ogram; and it admits of 

being split in three directions 
parallel to two of these op- 
posed faces. Even in such 
wise, if you will, that all the 
six faces are equal and similar 
rhombuses. The figure here 
added represents a piece of 
this Crystal. The obtuse 

angles of all the parallelo- "~p 

grams, as C, D, here, are angles of 101 degrees 52 minutes, 



and consequently the acute angles, such as A and B, are of 
78 degrees 8 minutes. 

5. Of the solid angles there are two opposite to one 
another, such as C and E, which are each composed of 
three equal obtuse plane angles. The other six are com- 
posed of two acute angles and one obtuse. All that I have 
just said has been likewise remarked by Mr. Bartholinus 
in the aforesaid treatise; if we differ it is only slightly about 
the values of the angles. He recounts moreover some other 
properties of this Crystal; to wit, that when rubbed 
against cloth it attracts straws and other light things as do 
amber, diamond, glass, and Spanish wax. Let a piece be 
covered with water for a' day or more, the surface loses its 
natural polish. When aquafortis is poured on it it produces 
ebullition, especially, as I have found, if the Crystal has 
been pulverized. I have also found by experiment that it 
may be heated to redness in the fire without being in any- 
wise altered or rendered less transparent; but a very violent 
fire calcines it nevertheless. Its transparency is scarcely 
less than that of water or of Rock Crystal, and devoid of 
colour. But rays of light pass through it in another fashion 
and produce those marvellous refractions the causes of which 
I am now going to try to explain ; reserving for the end of 
this Treatise the statement of my conje6lures touching the 
formation and extraordinary configuration of this Crystal. 

6. In all other transparent bodies that we know there 
is but one sole and simple refraction ; but in this substance 
there are two different ones. The effeft is that obje6ts seen 
through it, especially such as are placed right against it, 
appear double; and that a ray of sunlight, falling on one 
of its surfaces, parts itself into two rays and traverses the 
Crystal thus. 

7- ^ 


7. It is again a general law in all other transparent bodies 
that the ray which falls perpendicularly on their surface 
passes straight on without suffering refraction, and that an 
oblique ray is always refrafted. But in this Crystal the 
perpendicular ray suffers refra&ion, and there are oblique 
rays which pass through it quite straight. 

8. But in order to explain these phenomena more par- 
ticularly, let there be, in the first place, a piece ABFE of 
the same Crystal, and let the obtuse angle ACB, one of 
the three which constitute the equilateral solid angle C, 
be divided into two equal parts by the straight line CG, 
and let it be conceived that the Crystal is intersected by 
a plane which passes through this line and through the 
side CF, which plane will necessarily be perpendicular to 



the surface AB ; and its se<5tion in the Crystal will form a 
parallelogram GCFH. We will call this sedtion the 
principal section of the Crystal. 

9. Now if one covers the surface AB, leaving there 
only a small aperture at the point K, situated in the 
straight line CG, and if one exposes it to the sun, so that 
his rays face it perpendicularly above, then the ray IK 
will divide itself at the point K into two, one of which 
will continue to go on straight by KL, and the other will 
separate itself along the straight line KM, which is in 
the plane GCFH, and which makes with KL an angle 
of about 6 degrees 40 minutes, tending from the side of 
the solid angle C ; and on emerging from the other side 
of the Crystal it will turn again parallel to IK, along MZ. 
And as, in this extraordinary refra6tion, the point M is 
seen by the refracted ray MKI, which I consider as 
going to the eye at I, it necessarily follows that the point 
L, by virtue of the same refraction, will be seen by the 
refra&ed ray LRI, so that LR will be parallel to MK if 
the distance from the eye KI is supposed very great. 
The point L appears then as being in the straight line 
IRS ; but the same point appears also, by ordinary re- 
fraftion, to be in the straight line IK, hence it is neces- 
sarily judged to be double. And similarly if L be a small 
hole in a sheet of paper or other substance which is laid 
against the Crystal, it will appear when turned towards 
daylight as if there were two holes, which will seem the 
wider apart from one another the greater the thickness 
of the Crystal. 

10. Again, if one turns the Crystal in such wise that 
an incident ray NO, of sunlight, which I suppose to be 
in the plane continued from GCFH, makes with GC an 



angle of 73 degrees and 20 minutes, and is consequently 
nearly parallel to the edge CF, which makes with FH an 
angle of 70 degrees 57 minutes, according to the calcula- 
tion which I shall put at the end, it will divide itself at 
the point O into two rays, one of which will continue 
along OP in a straight line with NO, and will similarly 
pass out of the other side of the crystal without any re- 
fraction; but the other will be refraCted and will go along 
OQ^ And it must be noted that it is special to the plane 
through GCF and to those which are parallel to it, that 
all incident rays which are in one of these planes continue 
to be in it after they have entered the Crystal and have 
become double ; for it is quite otherwise for rays in all 
other planes which intersect the Crystal, as we shall see 

1 1 . I recognized at first by these experiments and by 
some others that of the two refractions which the ray 
suffers in this Crystal, there is one which follows the 
ordinary rules ; and it is this to which the rays KL and OQ 
belong. This is why I have distinguished this ordinary 
refraction from the other; and having measured it by 
exaCt observation, I found that its proportion, considered 
as to the Sines of the angles which the incident and 
refra6ted rays make with the perpendicular, was very pre- 
cisely that of 5 to 3, as was found also by Mr. Bartholinus, 
and consequently much greater than that of Rock Crystal, 
or of glass, which is nearly 3 to 2. 

12. The mode of making these observations exaCtly is 
as follows. Upon a leaf of paper fixed on a thoroughly 
flat table there is traced a black line AB, and two 
others, CED and KML, which cut it at right angles 
and are more or less distant from one another according 

i as 


as it is desired to examine a ray that is more or less 
oblique. Then place the Crystal upon the intersedlion 
E so that the line AB concurs with that which bisects 
the obtuse angle of the lower surface, or with some 
line parallel to it. Then by placing the eye direftly 
above the line AB it will appear single only; and one 
will see that the portion viewed through jjthe Crystal 

and the portions which appear outside it, meet together 
in a straight line : but the line CD will appear double, 
and one can distinguish the image which is due to 
regular refraction by the circumstance that when one 
views it with both eyes it seems raised up more than the 
other, or again by the circumstance that, when the Crystal 
is turned around on the paper, this image remains station- 
ary, whereas the other image shifts and moves entirely 
around. Afterwards let the eye be placed at I (remaining 



always in the plane perpendicular through AB) so that it 
views the image which is formed by regular refraction of 
the line CD making a straight line with the remainder 
of that line which is outside the Crystal. And then, 
marking on the surface of the Crystal the point H where 
the intersection E appears, this point will be direCtly above 
E. Then draw back the eye towards O, keeping always 
in the plane perpendicular through AB, so that the image 
of the line CD, which is formed by ordinary refraCtion, 
may appear in a straight line with the line KL viewed 
without refraction ; and then mark on the Crystal the 
point N where the point of intersection E appears. 

13. Then one will know the length and position of 
the lines NH, EM, and of HE, which is the thickness 
of the Crystal : which lines being traced separately upon 
a plan, and then joining NE and NM which cuts HE at 
P, the proportion of the refraCtion will be that of EN to 
NP, because these lines are to one another as the sines of 
the angles NPH, NEP, which are equal to those which 
the incident ray ON and its refraCtion NE make with 
the perpendicular to the surface. This proportion, as I 
have said, is sufficiently precisely as 5 to 3, and is always 
the same for all inclinations of the incident ray. 

14. The same mode of observation has also served me 
for examining the extraordinary or irregular refraCtion of 
this Crystal. For, the point H having been found and 
marked, as aforesaid, direCtly above the point E, I observed 
the appearance of the line CD, which is made by the 
extraordinary refraCtion ; and having placed the eye at Q, 
so that this appearance made a straight line with the 
line KL viewed without refraCtion, I ascertained the 
triangles REH, RES, and consequently the angles RSH, 




RES, which the incident and the refra&ed ray make 
with the perpendicular. 

15. But I found in this refraftion that the ratio of 
FR to RS was not constant, like the ordinary refradtion, 
but that it varied with the varying obliquity of the in- 
cident ray. 

1 6. I found also that when QRE made a straight line, 
that is, when the incident ray entered the Crystal with- 
out being refradted (as I ascertained by the circumstance 
that then the point E viewed by the extraordinary refrac- 
tion appeared in the line CD, as seen without refradtion) 
I found, I say, then that the angle QRG was 73 degrees 
20 minutes, as has been already remarked ; and so it is 

not the ray parallel 
to the edge of 
the Crystal, which 
crosses it in a 
straight line with- 
as Mr. Bartholinus 
believed, since that 
inclination is only 
70 degrees 57 
minutes, as was 
stated above. And 
this is to be noted, 
"F in order that no 
one may search in 
vain for the cause 

of the singular property of this ray in its parallelism 

to the edges mentioned, 

17. Finally, continuing my observations to discover the 




nature of this refraCtion, I learned that it obeyed the 
following remarkable rule. Let the parallelogram GCFH, 
made by the principal seCtion of the Crystal, as previously 
determined, be traced separately. I found then that 
always, when the inclinations of two rays which come 
from opposite sides, as VK, SK here, are equal, their re- 
fraCtions KX and KT meet the bottom line HF in such 
wise that points X and T are equally distant from the point 
M, where the refraCtion of the perpendicular ray IK falls ; 
and this occurs also for refractions in other sections of this 
Crystal. But before speaking of those, which have also 
other particular properties, we will investigate the causes 
of the phenomena which I have already reported. 

It was after having explained the refraction of ordinary 
transparent bodies by means of the spherical emanations 
of light, as above, that I resumed my examination of the 
nature of this Crystal, wherein I had previously been 
unable to discover anything. 

1 8. As there were two different refractions, I conceived 
that there were also two different emanations of waves of 
light, and that one could occur in the ethereal matter ex- 
tending through the body of the Crystal. Which matter, 
being present in much larger quantity than is that of the 
particles which compose it, was alone capable of causing 
transparency, according to what has been explained here- 
tofore. I attributed to this emanation of waves the regular 
refraCtion which is observed in this stone, by supposing 
these waves to be ordinarily of spherical form, and having 
a slower progression within the Crystal than they have 
outside it; whence proceeds refraCtion as I have demon- 

19. As to the other emanation which should produce 



the irregular refra6tion, I wished to try what Elliptical 
waves, or rather spheroidal waves, would do ; and these I 
supposed would spread indifferently both in the ethereal 
matter diffused throughout the crystal and in the particles of 
which it is composed, according to the last mode in which 
I have explained transparency. It seemed to me that the 
disposition or regular arrangement of these particles could 
contribute to form spheroidal waves (nothing more being 
required for this than that the successive movement of 
light should spread a little more quickly in one direction 
than in the other) and I scarcely doubted that there were 
in this crystal such an arrangement of equal and similar 
particles, because of its figure and of its angles with their 
determinate and invariable measure. Touching which 
particles, and their form and disposition, I shall, at the end 
of this Treatise, propound my conjectures and some ex- 
periments which confirm them. 

20. The double emission of waves of light, which I had 
imagined, became more probable to me after I had observed 
a certain phenomenon in the ordinary [Rock] Crystal, 
which occurs in hexagonal form, and which, because of 
this regularity, seems also to be composed of particles, of 
definite figure, and ranged in order. This was, that this 
crystal, as well as that from Iceland, has a double refraction, 
though less evident. For having had cut from it some 
well polished Prisms of different sections, I remarked in all, 
in viewing through them the flame of a candle or the lead 
of window panes, that everything appeared double, though 
with images not very distant from one another. Whence 
I understood the reason why this substance, though so 
transparent, is useless for Telescopes, when they have ever 
so little length. 

21. Now 


2 1 . Now this double refraftion, according to my Theory 
hereinbefore established, seemed to demand a double emis- 
sion of waves of light, both of them spherical (for both the 
refractions are regular) and those of one series a little 
slower only than the others. For thus the phenomenon is 
quite naturally explained, by postulating substances which 
serve as vehicle for these waves, as I have done in the case 
of Iceland Crystal. I had then less trouble after that in ad- 
mitting two emissions of waves in one and the same body. 
And since it might have been objefted that in composing 
these two kinds of crystal of equal particles of a certain 
figure, regularly piled, the interstices which these particles 
leave and which contain the ethereal matter would scarcely 
suffice to transmit the waves of light which I have localized 
there, I removed this difficulty by regarding these particles 
as being of a very rare texture, or rather as composed of 
other much smaller particles, between which the ethereal 
matter passes quite freely. This, moreover, necessarily 
follows from that which has been already demonstrated 
touching the small quantity of matter of which the bodies 
are built up. 

22. Supposing then these spheroidal waves besides the 
spherical ones, I began to examine whether they could 
serve to explain the phenomena of the irregular refraction, 
and how by these same phenomena I could determine the 
figure and position of the spheroids : as to which I obtained 
at last the desired success, by proceeding as follows. 

23. I considered first the effeft of waves so formed, as 
respedts the ray which falls perpendicularly on the flat 
surface of a transparent body in which they should spread 
in this manner. I took AB for the exposed region of the 
surface. And, since a ray perpendicular to a plane, and 



S A 







coming from a very distant source of light, is nothing else, 
according to the precedent Theory, than the incidence of 

a portion of the wave 
parallel to that plane, I 
supposed the straight line 
RC, parallel and equal to 
AB, to be a portion of a 
wave of light, in which an 
infinitude of points such as 
RH/^C come to meet the 
surface AB at the points 
AKB. Then instead of 
the hemispherical partial 
waves which in a body of ordinary refraction would spread 
from each of these last points, as we have above explained 
in treating of refraftion, these must here be hemispheroids. 
The axes (or rather the major diameters) of these I supposed 
to be oblique to the plane AB, as is AV the semi-axis or 
semi-major diameter of the spheroid SVT, which represents 
the partial wave coming from the point A, after the wave 
RC has reached AB. I say axis or major diameter, because 
the same ellipse SVT may be considered as the se6lion of a 
spheroid of which the axis is AZ perpendicular to AV. But, 
for the present, without yet deciding one or other, we will 
consider these spheroids only in those sections of them 
which make ellipses in the plane of this figure. Now taking 
a certain space of time during which the wave SVT 
has spread from A, it would needs be that from all the 
other points K^B there should proceed, in the same time, 
waves similar to SVT and similarly situated. And the 
common tangent NQ of all these semi-ellipses would be 
the propagation of the wave RC which fell on AB, and 



would be the place where this movement occurs in much 
greater amount than anywhere else, being made up of arcs 
of an infinity of ellipses, the centres of which are along 
the line AB. 

24. Now it appeared that this common tangent NQ was 
parallel to AB, and of the same length, but that it was not 
directly opposite to it, since it was comprised between the 
lines AN, BQ, which are diameters of ellipses having A 
and B for centres, conjugate with respeft to diameters 
which are not in the straight line AB. And in this way 
I comprehended, a matter which had seemed to me very 
difficult, how a ray perpendicular to a surface could suffer 
refradtion on entering a transparent body; seeing that 
the wave RC, having come to the aperture AB, went 
on forward thence, spreading between the parallel lines 
AN, BQ, yet itself remaining always parallel to AB, so 
that here the light does not spread along lines perpen- 
dicular to its waves, as in ordinary refraftion, but along 
lines cutting the waves obliquely. 

25. Inquiring subsequently what might be the position 
and form of these spheroids in the crystal, I considered that 
all the six faces produced pre- 
cisely the same refra&ions. 

Taking, then, the parallele- 
piped AFB, of which the ob- 
tuse solid angle C is contained 
between the three equal plane 
angles, and imagining in it 
the three principal se&ions, 
one of which is perpendicular 
to the face DC and passes through the edge CF, another 
perpendicular to the face BF passing through the edge 

K CA, 


CA, and the third perpendicular to the face AF passing 
through the edge BC; I knew that the refra&ions of the 
incident rays belonging to these three planes were all 
similar. But there could be no position of the spheroid 
which would have the same relation to these three sections 
except that in which the axis was also the axis of the solid 
angle C. Consequently I saw that the axis of this angle, 
that is to say the straight line which traversed the crystal 
from the point C with equal inclination to the edges CF, 
CA, CB was the line which determined the position of 
the axis of all the spheroidal waves which one imagined 
to originate from some point, taken within or on the sur- 
face of the crystal, since all these spheroids ought to be 
alike, and have their axes parallel to one another. 

26. Considering after this the plane of one of these 
three se6lions, namely that through GCF, the angle of 
which is 109 degrees 3 minutes, since the angle F was 
shown above to be 70 degrees 57 minutes; and, imagining 
a spheroidal wave about the centre C, I knew, because 
I have just explained it, that its axis must be in the same 
plane, the half of which axis I have marked CS in the 
next figure: and seeking by calculation (which will be 
given with others at the end of this discourse) the value 
of the angle CGS, 1 found it 45 degrees 20 minutes. 

27. To know from this the form of this spheroid, that is 
to say the proportion of the semi-diameters CS, CP, of its 
elliptical section, which are perpendicular to one another, 
I considered that the point M where the ellipse is touched 
by the straight line FH, parallel to CG, ought to be so 
situated that CM makes with the perpendicular CL an 
angle of 6 degrees 40 minutes; since, this being so, this 
ellipse satisfies what has been said about the refra6lion of 



the ray perpendicular to the surface CG, which is inclined 
to the perpendicular CL by the same angle. This, then, 
being thus dis- 
posed, and tak- / 
ing CM at f 
100,000 parts, 
I found by the 
cal culat ion 
which will be 
given at the 
end, the semi- 
major dia- 
meter CP to 
be 105,032, 

and the semi-axis CS to be 93,410, the ratio of which 
numbers is very nearly 9 to 8; so that the spheroid was 
of the kind which resembles a compressed sphere, being 
generated by the revolution of an ellipse about its smaller 
diameter. I found also the value of CG the semi-diameter 
parallel to the tangent ML to be 98,779. 

28. Now passing to the investigation of the refractions 
which obliquely incident rays must undergo, according to 
our hypothesis of spheroidal waves, I saw that these re- 
fraCtions depended on the ratio between the velocity of 
movement of the light outside the crystal in the ether, 
and that within the crystal. For supposing, for example, 
this proportion to be such that while the light in the 
crystal forms the spheroid GSP, as I have just said, it 
forms outside a sphere the semi-diameter of which is equal 
to the line N which will be determined hereafter, the 
following is the way of finding the refraction of the in- 
cident rays. Let there be such a ray RC falling upon the 



surface CK. Make CO perpendicular to RC, and across 
the angle KCO adjust OK, equal to N and perpendicular 
to CO; then draw KI, which touches the Ellipse GSP, 
and from the point of contaft I join 1C, which will be 
the required refraftion of the ray RC. The demonstration 
of this is, it will be seen, entirely similar to that of which 
we made use in explaining ordinary refra6tion. For the 

refra&ion of the ray RC is nothing else than the progres- 
sion of the portion C of the wave CO, continued in the 
crystal. Now the portions H of this wave, during the 
time that O came to K, will have arrived at the surface 
CK along the straight lines H#, and will moreover have 
produced in the crystal around the centres x some hemi- 
spheroidal partial waves similar to the hemi-spheroidal 
GSPg, and similarly disposed, and of which the major 



and minor diameters will bear the same proportions to 
the lines xv (the continuations of the lines H# up to KB 
parallel to CO) that the diameters of the spheroid GSPg 
bear to the line CB, or N. And it is quite easy to see 
that the common tangent of all these spheroids, which 
are here represented by Ellipses, will be the straight line 
IK, which consequently will be the propagation of the 
wave CO; and the point I will be that of the point C, 
conformably with that which has been demonstrated in 
ordinary refraction. 

Now as to finding the point of contaCt I, it is known 
that one must find CD a third proportional to the lines 
CK, CG, and draw DI parallel to CM, previously deter- 
mined, which is the conjugate diameter to CG; for then, 
by drawing KI it touches the Ellipse at I. 

29. Now as we have found CI the refraction of the ray 
RC, similarly one will find C/ the refraCtion of the ray rC, 
which comes from the opposite side, by making Co perpen- 
dicular to rC and following out the rest of the construction 
as before. Whence one sees that if the ray rC is inclined 
equally with RC, the line Cd will necessarily be equal to 
CD, because Ck is equal to CK, and Cg to CG. And in 
consequence I/ will be cut at E into equal parts by the 
line CM, to which DI and di are parallel. And because 
CM is the conjugate diameter to CG, it follows that /I will 
be parallel to gG. Therefore if one prolongs the refraCted 
rays CI, C/, until they meet the tangent ML at T and /, 
the distances MT, M/, will also be equal. And so, by our 
hypothesis, we explain perfectly the phenomenon men- 
tioned above; to wit, that when there are two rays equally 
inclined, but coming from opposite sides, as here the rays 
RC, re, their refractions diverge equally from the line 



followed by the refra6lion of the ray perpendicular to the 
surface, by considering these divergences in the direftion 
parallel to the surface of the crystal. 

30. To find the length of the line N, in proportion to 
CP, CS, CG, it must be determined by observations of 
the irregular refraftion which occurs in this section of the 
crystal; and I find thus that the ratio of N to GC is 
just a little less than 8 to 5. And having regard to some 
other observations and phenomena of which I shall speak 
afterwards, I put N at 156,962 parts, of which the semi- 
diameter CG is found to contain 98,779, making this ratio 
8 to 5 T 1 T . Now this proportion, which there is between 
the line N and CG, may be called the Proportion of the 
Refraction; similarly as in glass that of 3 to 2, as will be 
manifest when I shall have explained a short process in 
the preceding way to find the irregular refractions. 

3 1 . Supposing then, in the next figure, as previously, the 
surface of the crystal gG, the Ellipse GPg, and the line 
N; and CM the refradlion of the perpendicular ray FC, 
from which it diverges by 6 degrees 40 minutes. Now let 
there be some other ray RC, the refra&ion of which must 
be found. 

About the centre C, with semi-diameter CG, let the 
circumference gRG be described, cutting the ray RC at 
R; and let RV be the perpendicular on CG. Then as the 
line N is to CG let CV be to CD, and let DI be drawn 
parallel to CM, cutting the Ellipse ^MG at I; then join- 
ing CI, this will be the required refraftion of the ray RC. 
Which is demonstrated thus. 

Let CO be perpendicular to CR, and across the angle 
OCG let OK be adjusted, equal to N and perpendicular to 
CO, and let there be drawn the straight line KI, which if it 



is demonstrated to be a tangent to the Ellipse at I, it will be 
evident by the things heretofore explained that CI is the 
refradtion of the ray RC. Now since the angle RCO is a 
right angle, it is easy to see that the right-angled triangles 
RCV, KCO, are similar. As then, CK is to KO, so also 


is RC to CV. But KO is equal to N, and RC to CG: 
then as CK is to N so will CG be to CV. But as N is to 
CG, so, by construdion, is CV to CD. Then as CK is 
to CG so is CG to CD. And because DI is parallel to 
CM, the conjugate diameter to CG, it follows that KI 
touches the Ellipse at I; which remained to be shown. 
32. One sees then that as there is in the refraction of 



ordinary media a certain constant proportion between the 
sines of the angles which the incident ray and the refra&ed 
ray make with the perpendicular, so here there is such a 
proportion between CV and CD or IE; that is to say be- 
tween the Sine of the angle which the incident ray makes 
with the perpendicular, and the horizontal intercept, in 
the Ellipse, between the refra6lion of this ray and the 
diameter CM, For the ratio of CV to CD is, as has been 
said, the same as that of N to the semi-diameter CG. 

33. I will add here, before passing away, that in com- 
paring together the regular and irregular refraction of 
this crystal, there is this remarkable fat, that if ABPS 
be the spheroid by which light spreads in the Crystal in 
a certain space of time (which spreading, as has been 
said, serves for the irregular refra&ion), then the inscribed 
sphere BVST is the extension in the same space of time of 
the light which serves for the regular refradlion. 

For we have stated before this, that the line N being 

the radius of a spherical wave of 
light in air, while in the crystal 
it spread through the spheroid 
ABPS, the ratio of N to CS will 
be 156,962 to 93,410. But it has 
also been stated that the proportion 
of the regular refraftion was 5 to 
3 ; that is to say, that N being the 
radius of a spherical wave of light 
in air, its extension in the crystal 
would, in the same space of time, 
form a sphere the radius of which 
would be to N as 3 to 5. Now 156,962 is to 93,410 as 
5 to 3 less ^y. So that it is sufficiently nearly, and may 





be exaftly, the sphere BVST, which the light describes 
for the regular refraction in the crystal, while it describes 
the spheroid BPSA for the irregular refraCtion, and while 
it describes the sphere of radius N in air outside the 

Although then there are, according to what we have 
supposed, two different propagations of light within the 
crystal, it appears that it is only in directions perpendi- 
cular to the axis BS of the spheroid that one of these 
propagations occurs more rapidly than the other; but 
that they have an equal velocity in the other direction, 
namely, in that parallel to the same axis BS, which is 
also the axis of 
the obtuse angle 
of the crystal. 

34. The pro- 
portion of the re- 
fraCtion being 
what we have just 
seen, I will now 
show that there 
necessarily follows 
thence that not- 
able property of 
the ray which fall- 
ing obliquely on 
the surface of the 
crystal enters it 
without suffering 
refraction. For 
supposing the same things as before, and that the ray 
RC makes with the same surface gG the angle RCG of 

L 73 degrees 


73 degrees 20 minutes, inclining to the same side as 
the crystal (of which ray mention has been made above) ; 
if one investigates, by the process above explained, the 
refraftion CI, one will find that it makes exaftly a straight 
line with RC, and that thus this ray is not deviated at all, 
conformably with experiment. This is proved as follows 
by calculation. 

CG or CR being, as precedently, 98,779 ; CM being 
100,000; and the angle RCV 73 degrees 20 minutes, 
CV will be 28,330. But because CI is the refraction of 
the ray RC, the proportion of CV to CD is 156,962 to 
98,779, namely, that of N to CG ; then CD is 17,828. 

Now the reftangle'gDC is to the square of DI as the 
square of CG is to the square of CM ; hence DI or CE 
will be 98,353. But as CE is to El, so will CM be to 
MT, which will then be 18,127. And being added to 
ML, which is 1 1,609 (namely the sine of the angle LCM, 
which is 6 degrees 40 minutes, taking CM 100,000 as 
radius) we get LT 27,936; and this is to LC 99,324 as 
CV to VR, that is to say, as 29,938, the tangent of the 
complement of the angle RCV, which is 73 degrees 
20 minutes, is to the radius of the Tables. Whence it 
appears that RCIT is a straight line ; which was to be 

35. Further it will be seen that the ray CI in emerging 
through the opposite surface of the crystal, ought to pass 
out quite straight, according to the following demonstra- 
tion, which proves that the reciprocal relation of refrac- 
tion obtains in this crystal the same as in other trans- 
parent bodies ; that is to say, that if a ray RC in meeting 
the surface of the crystal CG is refracted as CI, the ray 
CI emerging through the opposite parallel surface of the 



crystal, which I suppose to be IB, will have its refraftion 
IA parallel to 
the ray RC. 

Let the same 
things be sup- 
posed as before ; 
that is to say, 
let CO, perpen- 
dicular to CR, 
represent a por- 
tion of a wave 
the continua- 
tion of which 
in the crystal is 
IK, so that the 
piece C will be 
continued on along the straight line CI, while O comes 
to K. Now if one takes a second period of time equal 
to the first, the piece K of the wave IK will, in this 
second period, have advanced along the straight line KB, 
equal and parallel to CI, because every piece of the 
wave CO, on arriving at the surface CK, ought to go 
on in the crystal the same as the piece C ; and in this 
same time there will be formed in the air from the point I 
a partial spherical wave having a semi-diameter IA equal 
to KO, since KO has been traversed in an equal time. 
Similarly, if one considers some other point of the wave 
IK, such as /z, it will go along hm^ parallel to CI, to meet 
the surface IB, while the point K traverses K/ equal to hm ; 
and while this accomplishes the remainder /B, there will 
start from the point m a partial wave the semi-diameter 
of which, mn, will have the same ratio to /B as IA to 



KB. Whence it is evident that this wave or semi-diameter 
mn, and the other of semi-diameter IA will have the same 
tangent BA. And similarly for all the partial spherical 
waves which will be formed outside the crystal by the 
impa6l of all the points of the wave IK against the surface 
of the Ether IB. It is then precisely the tangent BA 
which will be the continuation of the wave IK, outside 
the crystal, when the piece K has reached B. And in 
consequence IA, which is perpendicular to BA, will be 
the refra6lion of the ray CI on emerging from the crystal. 
Now it is clear that IA is parallel to the incident ray 
RC, since IB is equal to CK, and I A equal to KO, and 
the angles A and O are right angles. 

It is seen then that, according to our hypothesis, the 
reciprocal relation of refradtion holds good in this crystal 
as well as in ordinary transparent bodies ; as is thus in fa<5t 
found by observation. 

36. I pass now to the consideration of other sections of 
the crystal, and of the refraftions there produced, on 
which, as will be seen, some other very remarkable phe- 
nomena depend. 

Let ABH be a parallelepiped of crystal, and let the top 
surface AEHF be a perfect rhombus, the obtuse angles of 
which are equally divided by the straight line EF, and 
the acute angles by the straight line AH perpendicular 

The se6lion which we have hitherto considered is that 
which passes through the lines EF, EB, and which at the 
same time cuts the plane AEHF at right angles. Refrac- 
tions in this sedtion have this in common with the re- 
fradtions in ordinary media that the plane which is drawn 
through the incident ray and which also intersefts the 



surface of the crystal at right angles, is that in which the 
refra&ed ray also is found. But the refraftions which 
appertain to every other seftion of this crystal have this 
strange property that the refra&ed ray always quits the 
plane of the incident ray perpendicular to the surface, and 
turns away towards the side of the slope of the crystal. For 


which fa<5l we shall show the reason, in the first place, for 
the seftion through AH ; and we shall show at the same 
time how one can determine the refra6lion, according to 
our hypothesis. Let there be, then, in the plane which 
passes through AH, and which is perpendicular to the 
plane AFHE, the incident ray RC; it is required to find 
its refraction in the crystal. 

37. About 


37. About the centre C, which I suppose to be in the 
intersection of AH and FE, let there be imagined a 
hemi-spheroid QG^g-M, such as the light would form in 
spreading in the crystal, and let its section by the plane 
AEHF form the Ellipse QG^g-, the major diameter of 
which Q^, which is in the line AH, will necessarily be 
one of the major diameters of the spheroid; because the 
axis of the spheroid being in the plane through FEE, to 
which QC is perpendicular, it follows that QC is also 
perpendicular to the axis of the spheroid, and consequently 
QCg one of its major diameters. But the minor diameter 
of this Ellipse, Gg-, will bear to Qg the proportion which 
has been defined previously, Article 27, between CG and 
the major semi-diameter of the spheroid, CP, namely, that 
of 98,779 to 105,032. 

Let the line N be the length of the travel of light 
in air during the time in which, within the crystal, it 
makes, from the centre C, the spheroid QGggM. Then 
having drawn CO perpendicular to the ray CR and 
situate in the plane through CR and AH, let there be 
adjusted, across the angle AGO, the straight line OK 
equal to N and perpendicular to CO, and let it meet the 
straight line AH at K. Supposing consequently that CL 
is perpendicular to the surface of the crystal AEHF, and 
that CM is the refra&ion of the ray which falls perpendi- 
cularly on this same surface, let there be drawn a plane 
through the line CM and through KCH, making in the 
spheroid the semi-ellipse QM^, which will be given, 
since the angle MCL is given of value 6 degrees 40 
minutes. And it is certain, according to what has been 
explained above, Article 27, that a plane which would 
touch the spheroid at the point M, where I suppose the 



straight line CM to meet the surface, would be parallel 
to the plane QG^. If then through the point K one now 
draws KS parallel to Gg, which will be parallel also to 
QX, the tangent to the Ellipse QGq at Q; and if one 
conceives a plane passing through KS and touching the 
spheroid, the point of contact will necessarily be in the 
Ellipse QM^, because this plane through KS, as well as 
the plane which touches the spheroid at the point M, are 
parallel to QX, the tangent of the spheroid : for this 
consequence will be demonstrated at the end of this 
Treatise. Let this point of conta6t be at I, then making 
KC, QC, DC proportionals, draw DI parallel to CM; 
also join CL I say that CI will be the required refraction 
of the ray RC. This will be manifest if, in considering 
CO, which is perpendicular to the ray RC, as a portion 
of the wave of light, we can demonstrate that the con- 
tinuation of its piece C will be found in the crystal at I, 
when O has arrived at K. 

38. Now as in the Chapter on Reflexion, in demon- 
strating that the incident and reflected rays are always 
in the same plane perpendicular to the reflecting surface, 
we considered the breadth of the wave of light, so, 
similarly, we must here consider the breadth of the wave 
CO in the diameter Gg. Taking then the breadth Cc on 
the side toward the angle E, let the parallelogram COoc 
be taken as a portion of a wave, and let us complete the 
parallelograms CK&:, CI/r, KI/, QK&o. In the time 
then that the line Oo arrives at the surface of the crystal 
at K, all the points of the wave COoc will have arrived 
at the re6tangle Kr along lines parallel to OK ; and from 
the points of their incidences there will originate, beyond 
that, in the crystal partial hemi-spheroids, similar to the 



hemi-spheroid QM^, and similarly disposed. These hemi- 
spheroids will necessarily all touch the plane of the 
parallelogram KI/> at the same instant that Oo has 
reached K. Which is easy to comprehend, since, of 
these hemi-spheroids, all those which have their centres 
along the line CK, touch this plane in the line KI (for 
this is to be shown in the same way as we have demon- 
strated the refra6tion of the oblique ray in the principal 
se6tion through EF) and all those which have their 
centres in the line Cc will touch the same plane KI in the 
line I/; all these being similar to the hemi-spheroid 
QMy. Since then the parallelogram K/ is that which 
touches all these sphe'roids, this same parallelogram will 
be precisely the continuation of the wave COoc in the 
crystal, when Oo has arrived at K, because it forms the 
termination of the movement and because of the quantity 
of movement which occurs more there than anywhere 
else : and thus it appears that the piece C of the wave 
COoc has its continuation at I ; that is to say, that the 
ray RC is refracted as CL 

From this it is to be noted that the proportion of the 
refraction for this se&ion of the crystal is that of the line 
N to the semi-diameter CQ; by which one will easily 
find the refradtions of all incment rays, in the same way 
as we have shown previously for the case of the section 
through FE; and the demonstration will be the same. 
But it appears that the said proportion of the refraftion 
is less here than in the section through FEB ; for it was 
there the same as the ratio of N to CG, that is to say, as 
156,962 to 98,779, very nearly as 8 to 5 ; and here it is 
the ratio of N to CQj:he major semi-diameter of the 
spheroid, that is to say, as 156,962 to 105,032, very nearly 



as 3 to 2, but just a little less. Which still agrees perfectly 
with what one finds by observation. 

39. For the rest, this diversity of proportion of refraction 
produces a very singular effeft in this Crystal ; which is 
that when it is placed upon a sheet of paper on which 
there are letters or anything else marked, if one views it 
from above with the two eyes situated in the plane of the 
section through EF, one sees the letters raised up by this 
irregular refraftion more than when one puts one's eyes 
in the plane of section through AH : and the difference of 
these elevations appears by comparison with the other 
ordinary refra6tion of the crystal, the proportion of which is 
as 5 to 3 5 an d which always raises the letters equally, and 
higher than the irregular refra&ion does. For one sees 
the letters and the paper on which they are written, as 
on two different stages at the same time ; and in the first 
position of the eyes, namely, when they are in the plane 
through AH these two stages are four times more distant 
from one another than when the eyes are in the plane 
through EF. 

We will show that this effeft follows from the refrac- 
tions; and it will enable us at the same time to ascertain 
the apparent place of a point of an objet placed immedi- 
ately under the crystal, according to the different situation 
of the eyes. 

40. Let us see first by how much the irregular refraftion 
of the plane through AH ought to lift the bottom of the 
crystal. Let the plane of this figure represent separately 
the section through Qg and CL, in which seflion there is 
also the ray RC, and let the semi-elliptic plane through 
Qq and CM be inclined to the former, as previously, by an 
angle of 6 degrees 40 minutes ; and in this plane CI is 
then the refralion of the ray RC. 

M If 



If now one considers the point I as at the bottom of 
the crystal, and that it is viewed by the rays ICR, Icr, 

refradted equally at 
the points Cr, 
which should be 
equally distant 
from D, and that 
these rays meet the 
two eyes at Rr; it 
is certain that the 
\q point I will appear 
raised to S where 
the straight lines 
RC, rr, meet; 
which point S is 
in DP, perpendicu- 
lar to Qg. And if 
upon DP there is drawn the perpendicular IP, which 
will lie at the bottom of the crystal, the length SP will 
be the apparent elevation of the point I above the bottom. 
Let there be described on Q^ a semicircle cutting the 
ray CR at B, from which BV is drawn perpendicular to 
Qy; and let the proportion of the refraction for this section 
be, as before, that of the line N to the semi-diameter CQ. 
Then as N is to CQj>o is VC to CD, as appears by the 
method of finding the refradtion which we have shown 
above, Article 31; but as VC is to CD, so is VBto DS. Then 
as N is to CQ, so is VB to DS. Let ML be perpendicular to 
CL. And because I suppose the eyes Rr to be distant about 
a foot or so from the crystal, and consequently the angle 
RSr very small, VB may be considered as equal to the 
semi-diameter CQ, and DP as equal to CL; then as N is to 



CQ so is CQ to DS. But N is valued at 156,962 parts, of 
which CM contains 100,000 and CQ 105,032. Then DS 
will have 70,283. But CL is 99,324, being the sine of 
the complement of the angle MCL which is 6 degrees 40 
minutes; CM being supposed as radius. Then DP, con- 
sidered as equal to CL, will be to DS as 99,324 to 70,283. 
And so the elevation of the point I by the refraftion of 
this section is known. 

41. Now let there be represented the other seftion 
through EF in the figure before the preceding one ; and 
let CMg be the semi-ellipse, considered in Articles 27 and 
28, which is made by cutting a spheroidal wave having 
centre C. Let the point I, taken in this ellipse, be imagined 
again at the bottom of the Crystal; and let it be viewed 
by the refrafted rays ICR, Icr, which go to the two eyes; 
CR and cr being 
equally inclined to the 
surface of the crystal 
Gg. This being so, if 
one draws ID parallel 
to CM, which I sup- 
pose to be the refra6tion 
of the perpendicular 
ray incident at the . 
point C, the distances 
DC, DC, will be equal, 
as is easy to see by that 
which has been demon- 
strated in Article 28. 
Now it is certain that 
the point I should appear at S where the straight lines RC, 
re, meet when prolonged; and that this point will fall in the 



line DP perpendicular to Gg. If one draws IP perpendicu- 
lar to this DP, it will be the distance PS which will 
mark the apparent elevation of the point I. Let there be 
described on Gg a semicircle cutting CR at B, from 
which let BV be drawn perpendicular to Gg; and let N 
to GC be the proportion of the refraftion in this seftion, 
as in Article 28. Since then CI is the refraftion of the 
radius BC, and DI is parallel to CM, VC must be to CD as 
N to GC, according to what has been demonstrated in 
Article 31. But as VC is to CD so is BV to DS. Let 
ML be drawn perpendicular to CL. And because I con- 
sider, again, the eyes to be distant above the crystal, BV 
is deemed equal to th6 semi-diameter CG; and hence DS 
will be a third proportional to the lines N and CG: also 
DP will be deemed equal to CL. Now CG consisting of 
98,778 parts, of which CM contains 100,000, N is taken 
as 156,962. Then DS will be 62,163. But CL is also 
determined, and contains 99,324 parts, as has been said in 
Articles 34 and 40. Then the ratio of PD to DS will be 
as 99,324 to 62,163. And thus one knows 
the elevation of the point at the bottom I by 
the refradtion of this section; and it appears 
that this elevation is greater than that by 
the refra&ion of the preceding se6lion, since 
the ratio of PD to DS was there as 99,324 
to 70,283. 

But by the regular refra&ion of the 
crystal, of which we have above said that 
the proportion is 5 to 3, the elevation of 
the point I, or P, from the bottom, will be 
of the height DP; as appears by this figure, where the point 
P being viewed by the rays PCR, Per, refrafted equally 



at the surface O, this point must needs appear to be at S, 
in the perpendicular PD where the lines RC, re, meet 
when prolonged: and one knows that the line PC is to 
CS as 5 to 3, since they aVe to one another as the sine of the 
angle CSP or DSC is to the sine of the angle SPC. And 
because the ratio of PD to DS is deemed the same as that 
of PC to CS, the two eyes Rr being supposed very far above 
the crystal, the elevation PS will thus be -f- of PD. 

42. If one takes a straight line AB for the thickness of 
the crystal, its point B being at the bottom, and 

if one divides it at the points C, D, E, according 
to the proportions of the elevations found, making 
AE f of AB, AB to AC as 99,324 to 70,283, and 
AB to AD as 99,324 to 62,163, these points will 
divide AB as in this figure. And it will be found 
that this agrees perfectly with experiment; that is , 
to say by placing the eyes above in the plane which 
cuts the crystal according to the shorter diameter 
of the rhombus, the regular refraction will lift up 
the letters to E; and one will see the bottom, and 
the letters over which it is placed, lifted up to D 
by the irregular refraCtion. But by placing the eyes above 
in the plane which cuts the crystal according to the 
longer diameter of the rhombus, the regular refraCtion 
will lift the letters to E as before; but the irregular re- 
fra6lion will make them, at the same time, appear lifted 
up only to C; and in such a way that the interval CE 
will be quadruple the interval ED, which one previously 

43. I have only to make the remark here that in both the 
positions of the eyes the images caused by the irregular 
refra6tion do not appear direCtly below those which pro- 



ceed from the regular refraCtion, but they are separated 
from them by being more distant from the equilateral 
solid angle of the Crystal. That follows, indeed, from all 
that has been hitherto demonstrated about the irregular 
refraCtion; and it is particularly shown by these last demon- 
strations, from which one sees that the point I appears by 
irregular refraction at S in the perpendicular line DP, in 
which line also the image of the point P ought to appear by 
regular refraction, but not the image of the point I, which 
will be almost direCtly above the same point, and higher 
than S. 

But as to the apparent elevation of the point I in other 
positions of the eyes "above the crystal, besides the two 
positions which we have just examined, the image of that 
point by the irregular refraction will always appear be- 
tween the two heights of D and C, passing from one to 
the other as one turns one's self around about the immov- 
able crystal, while looking down from above. And all 
this is still found conformable to our hypothesis, as any one 
can assure himself after I shall have shown here the way 
of finding the irregular refractions which appear in all 
other sections of the crystal, besides the two which we 
have considered. Let us suppose one of the faces of the 
crystal, in which let there be the Ellipse HDE, the centre 
C of which is also the centre of the spheroid HME in which 
the light spreads, and of which the said Ellipse is the section. 
And let the incident ray be RC, the refraction of which it 
is required to find. 

Let there be taken a plane passing through the ray RC 
and which is perpendicular to the plane of the ellipse HDE, 
cutting it along the straight line BCK; and having in the 
same plane through RC made CO perpendicular to CR, 




let OK be adjusted across the angle OCK, so as to be 
perpendicular to OC and equal to the line N, which I sup- 
pose to measure 
the travel of the 
light in air during 
the time that it 
spreads in the 
crystal through 
the spheroid 
HDEM. Then in 
the plane of the 
Ellipse HDE let 
KT be drawn, 
through the point 
K, perpendicular 
to BCK. Now if one conceives a plane drawn through the 
straight line KT and touching the spheroid HME at I, 
the straight line CI will be the refraction of the ray RC, 
as is easy to deduce from that which has been demon- 
strated in Article 36. 

But it must be shown how one can determine the 
point of contadl I. Let there be drawn parallel to the 
line KT a line HF which touches the Ellipse HDE, 
and let this point of contact be at H. And having drawn 
a straight line along CH to meet KT at T, let there be 
imagined a plane passing through the same CH and through 
CM (which I suppose to be the refraction of the perpendicu- 
lar ray), which makes in the spheroid the elliptical section 
HME. It is certain that the plane which will pass through 
the straight line KT, and which will touch the spheroid, 
will touch it at a point in the Ellipse HME, according to 
the Lemma which will be demonstrated at the end of the 



Chapter. Now this point is necessarily the point I which 
is sought, since the plane drawn through TK can touch 
the spheroid at one point only. And this point I is easy 
to determine, since it is needful only to draw from the 
point T, which is in the plane of this Ellipse, the tangent 
TI, in the way shown previously. For the Ellipse HME 
is given, and its conjugate semi-diameters are CH and 
CM; because a straight line drawn through M, parallel 
to HE, touches the Ellipse HME, as follows from the 
fa6t that a plane taken through M, and parallel to the 
plane HDE, touches the spheroid at that point M, as is 
seen from Articles 27 and 23. For the rest, the position 
of this ellipse, with respe6t to the plane through the ray 
RC and through CK, is also given; from which it will be 
easy to find the position of CI, the refraction corresponding 
to the ray RC. 

Now it must be noted that the same ellipse HME 
serves to find the refra6tions of any other ray which may 
be in the plane through RC and CK. Because every 
plane, parallel to the straight line HF, or TK, which will 
touch the spheroid, will touch it in this ellipse, according 
to the Lemma quoted a little before. 

I have investigated thus, in minute detail, the properties 
of the irregular refra6lion of this Crystal, in order to see 
whether each phenomenon that is deduced from our hypo- 
thesis accords with that which is observed in fadt. And 
this being so it affords no slight proof of the truth of our 
suppositions and principles. But what I am going to add 
here confirms them again marvellously. It is this: that 
there are different sections of this Crystal, the surfaces 
of which, thereby produced, give rise to refradtions pre- 
cisely such as they ought to be, and as I had foreseen them, 
according to the preceding Theory. 



In order to explain what these seftions are, let ABKF 
be the principal se<5tion through the axis of the crystal 
ACK, in which there will also be the axis SS of a sphe- 
roidal wave of light 
spreading in the crystal 
from the centre C ; and 
the straight line which 
cuts SS through the 
middle and at right 
angles, namely PP, will 
be one of the major dia- 

Now as in the natural 
se6tion of the crystal, made by a plane parallel to two 
opposite faces, which plane is here represented by the 
line GG, the refra6lion of the surfaces which are pro- 
duced by it will be governed by the hemi-spheroids 
GNG, according to what has been explained in the pre- 
ceding Theory. Similarly, cutting the Crystal through 
NN, by a plane perpendicular to the parallelogram 
ABKF, the refraction of the surfaces will be governed by 
the hemi-spheroids NGN. And if one cuts it through PP, 
perpendicularly to the said parallelogram, the refraction 
of the surfaces ought to be governed by the hemi-spheroids 
PSP, and so for others. But I saw that if the plane NN 
was almost perpendicular to the plane GG, making the 
angle NCG, which is on the side A, an angle of 90 degrees 
40 minutes, the hemi-spheroids NGN would become similar 
to the hemi-spheroids GNG, since the planes NN and GG 
were equally inclined by an angle of 45 degrees 20 minutes 
to the axis SS. In consequence it must needs be, if our 
theory is true, that the surfaces which the seftion through 



NN produces should effe6t the same refractions as the sur- 
faces of the section through GG. And not only the surfaces 
of the sedtion NN but all other sections produced by planes 
which might be inclined to the axis at an angle equal to 
45 degrees 20 minutes. So that there are an infinitude of 
planes which ought to produce precisely the same re- 
fractions as the natural surfaces of the crystal, or as the 
sedtion parallel to any one of those surfaces which are 
made by cleavage. 

I saw also that by cutting it by a plane taken through 
PP, and perpendicular to the axis SS, the refradtion of the 
surfaces aught to be such that the perpendicular ray should 
suffer thereby no deviation; and that for oblique rays there 
would always be an irregular refraction, differing from the 
regular, and by which objedts placed beneath the crystal 
would be less elevated than by that other refradtion. 

That, similarly, by cutting the crystal by any plane 
through the axis SS, such as the plane of the figure is, the 
perpendicular ray ought to suffer no refradtion ; and that 
for oblique rays there were different measures for the 
irregular refradtion according to the situation of the plane 
in which the incident ray was. 

Now these things were found in fadt so; and, after 
that, I could not doubt that a similar success could be met 
with everywhere. Whence I concluded that one might 
form from this crystal solids similar to those which are its 
natural forms, which should produce, at all their surfaces, 
the same regular and irregular refradtions as the natural 
surfaces, and which nevertheless would cleave in quite other 
ways, and not in directions parallel to any of their faces. That 
out of it one would be able to fashion pyramids, having their 
base square, pentagonal, hexagonal, or with as many sides 



as one desired, all the surfaces of which should have the 
same refra6lions as the natural surfaces of the crystal, except 
the base, which will not refraft the perpendicular ray. 
These surfaces will each make an angle of 45 degrees 
20 minutes with the axis of the crystal, and the base will 
be the se6lion perpendicular to the axis. 

That, finally, one could also fashion out of it triangular 
prisms, or prisms with as many sides as one would, of which 
neither the sides nor the bases would refraft the perpen- 
dicular ray, although they would yet all cause double re- 
fra6lion for oblique rays. The cube is included amongst 
these prisms, the bases of which are sections perpendicular 
to the axis of the crystal, and the sides are sections parallel 
to the same axis. 

From all this it further appears that it is not at all in 
the disposition of the layers of which this crystal seems to 
be composed, and according to which it splits in three 
different senses, that the cause resides of its irregular re- 
fradlion ; and that it would be in vain to wish to seek it 

But in order that any one who has some of this stone 
may be able to find, by his own experience, the truth of 
what I have just advanced, I will state here the process of 
which I have made use to cut it, and to polish it. Cutting 
is easy by the slicing wheels of lapidaries, or in the way in 
which marble is sawn : but polishing is very difficult, and 
by employing the ordinary means one more often depolishes 
the surfaces than makes them lucent. 

After many trials, I have at last found that for this service 
no plate of metal must be used, but a piece of mirror glass 
made matt and depolished. Upon this, with fine sand and 
water, one smoothes the crystal little by little, in the same 



way as spe6lacle glasses, and polishes it simply by continu- 
ing the work, but ever reducing the material. I have not, 
however, been able to give it perfect clarity and trans- 
parency ; but the evenness which the surfaces acquire en- 
ables one to observe in them the effeCts of refraCtion better 
than in those made by cleaving the stone, which always 
have some inequality. 

Even when the surface is only moderately smoothed, if 
one rubs it over with a little oil or white of egg, it becomes 
quite transparent, so that the refraction is discerned in it 
quite distindtly. And this aid is specially necessary when 
it is wished to polish the natural surfaces to remove the in- 
equalities ; because one cannot render them lucent equally 
with the surfaces of other sections, which take a polish so 
much the better the less nearly they approximate to these 
natural planes. 

Before finishing the treatise on this Crystal, I will add 
one more marvellous phenomenon which I discovered after 
having written all the foregoing. For though I have not 
been able till now to find its cause, I do not for that rea- 
son wish to desist from describing it, in order to give op- 
portunity to others to investigate it. It seems that it will be 
necessary to make still further suppositions besides those 
which I have made ; but these will not for all that cease to 
keep their probability after having been confirmed by so 
many tests. 

The phenomenon is, that by taking two pieces of this 
crystal and applying them one over the other, or rather 
holding them with a space between the two, if all the 
sides of one are parallel to those of the other, then a ray 
of light, such as AB, is divided into two in the first piece, 
namely into BD and BC, following the two refractions, 



regular and irregular. On penetrating thence into the 
other piece each ray will pass there without further divid- 

ing itself in two ; but that one which underwent the regular 
refra6tion, as here DG, will undergo again only a regular 
refraCtion at GH ; and the other, CE, an irregular re- 
fraftion at EF. And the same thing occurs not only in this 
disposition, but also in all those cases in which the principal 
seCtion of each of the pieces is situated in one and the 
same plane, without it being needful for the two neigh- 
bouring surfaces to be parallel. Now it is marvellous why 
the rays CE and DG, incident from the air on the lower 
crystal, do not divide themselves the same as the first ray 
AB. One would say that it must be that the ray DG in 
passing through the upper piece has lost something which 
is necessary to move the matter which serves for the irre- 
gular refraCtion ; and that likewise CE has lost that which 



was necessary to move the matter whicn serves for regular 
refraction : but there is yet another thing which upsets 
this reasoning. It is that when one disposes the two crystals 
in such a way that the planes which constitute the principal 
sections interseft one another at right angles, whether the 
neighbouring surfaces are parallel or not, then the ray 
which has come by the regular refra&ion, as DG, under- 
goes only an irregular refraction in the lower piece ; and 
on the contrary the ray which has come by the irregular 
refraftion, as CE, undergoes only a regular refradtion. 

But in all the infinite other positions, besides those which 
I have just stated, the rays DG, CE, divide themselves 
anew each one into two, by refraction in the lower crystal, 
so that from the single ray AB there are four, sometimes 
of equal brightness, sometimes some much less bright 
than others, according to the varying agreement in the 
positions of the crystals : but they do not appear to have 
all together more light than the single ray AB. 

When one considers here how, while the rays CE, DG, 
remain the same, it depends on the position that one gives 
to the lower piece, whether it divides them both in two, or 
whether it does not divide them, and yet how the ray AB 
above is always divided, it seems that one is obliged to con- 
clude that the waves of light, after having passed through the 
first crystal, acquire a certain form or disposition in virtue 
of which, when meeting the texture of the second crystal, 
in certain positions, they can move the two different kinds 
of matter which serve for the two species of refraftion ; 
and when meeting the second crystal in another position 
are able to move only one of these kinds of matter. But 
to tell how this occurs, I have hitherto found nothing 
which satisfies me. 



Leaving then to others this research, I pass to what I 
have to say touching the cause of the extraordinary figure 
Df this crystal, and why it cleaves easily in three different 
senses, parallel to any one of its surfaces. 

There are many bodies, vegetable, mineral, and congealed 
salts, which are formed with certain regular angles and 
figures. Thus among flowers there are many which have 
their leaves disposed in ordered polygons, to the number 
of 3, 4, 5, or 6 sides, but not more. This well deserves to 
be investigated, both as to the polygonal figure, and as to 
why it does not exceed the number 6. 

Rock Crystal grows ordinarily in hexagonal bars, and 
diamonds are found which occur with a square point and 
polished surfaces. There is a species of small flat stones, 
piled up diredtly upon one another, which are all of pen- 
tagonal figure with rounded angles, and the sides a little 
folded inwards. The grains of gray salt which are formed 
from sea water affe<5t the figure, or at least the angle, of 
the cube ; and in the congelations of other salts, and in 
that of sugar, there are found other solid angles with per- 
fe6lly flat faces. Small snowflakes almost always fall in 
little stars with 6 points, and sometimes in hexagons with 
straight sides. And I have often observed, in water which 
is beginning to freeze, a kind of flat and thin foliage of ice, 
the middle ray of which throws out branches inclined at an 
angle of 60 degrees. All these things are worthy of being 
carefully investigated to ascertain how and by what artifice 
nature there operates. But it is not now my intention to treat 
fully of this matter. It seems that in general the regularity 
which occurs in these produ6lions comes from the arrange- 
ment of the small invisible equal particles of which they 
are composed. And, coming to our Iceland Crystal, I say 



that if there were a pyramid such as ABCD, composed of 
small rounded corpuscles, not spherical but flattened spher- 
oids, such as would be made by the 
rotation of the ellipse GH around its 
lesser diameter EF (of which the 
ratio to the greater diameter is very 
nearly that of i to the square root 
of 8) I say that then the solid angle 
of the point D would be equal to the 
obtuse and equilateral angle of this 
Crystal. I say, further, that if these 
corpuscles were lightly stuck together, 
oft breaking this pyramid it would 
break along faces parallel to those 
that make its point: and by this 
means, as it is easy to see, it would produce prisms simi- 
lar to those of the same crystal as this other figure re- 
presents. The reason is that when broken in this fashion 
a whole layer separates easily from its neighbouring layer 
since each spheroid has to be detached only from the three 
spheroids of the next layer ; of which three there is but one 
which touches it on its flattened surface, and the other two 
at the edges. And the reason why the surfaces separate 
sharp and polished is that if any spheroid of the neighbour- 
ing surface would come out by attaching itself to the surface 
which is being separated, it would be needful for it to de- 
tach itself from six other spheroids which hold it locked, 
and four of which press it by these flattened surfaces. Since 
then not only the angles of our crystal but also the manner 
in which it splits agree precisely with what is observed in 
the assemblage composed of such spheroids, there is great 
reason to believe that the particles are shaped and ranged 
in the same way. 




There is even probability enough that the prisms of this 
crystal are produced by the breaking up of pyramids, since 
Mr. Bartholinus relates that he occasionally found some 
pieces of triangularly pyramidal figure. But when a mass 
is composed interiorly only of these little spheroids thus 
piled up, whatever form it may have exteriorly, it is 
certain, by the same reasoning which I have just explained, 
that if broken it would produce similar prisms. It remains 
to be seen whether there are other reasons which confirm 
our conjecture, and 
whether there are none 
which are repugnant 
to it. 

It may be objedted 
that this crystal, being 
so composed, might be 
capable of cleavage in 
yet two more fashions; 
one of which would be 
along planes parallel to the base of the pyramid, that is 
to say to the triangle ABC; the other would be parallel 
to a plane the trace of which is marked by the lines GH, 
HK, KL. To which I say that both the one and the 
other, though practicable, are more difficult than those 
which were parallel to any one of the three planes of 
the pyramid; and that therefore, when striking on the 
crystal in order to break it, it ought always to split 
rather along these three planes than along the two others. 
When one has a number of spheroids of the form above 
described, and ranges them in a pyramid, one sees why the 
two methods of division are more difficult. For in the 
case of that division which would be parallel to the base, 

o each 


each spheroid would be obliged to detach itself from three 
others which it touches upon their flattened surfaces, which 
hold more strongly than the conta&s at the edges. And be- 
sides that, this division will not occur along entire layers, 
because each of the spheroids of a layer is scarcely held at 
all by the 6 of the same layer that surround it, since they 
only touch it at the edges; so that it adheres readily to 
the neighbouring layer, and the others to it, for the same 
reason; and this causes uneven surfaces. Also one sees by 
experiment that when grinding down the crystal on a 
rather rough stone, diretly on the equilateral solid angle, 
one verily finds much facility in reducing it in this direction, 
but much difficulty afterwards in polishing the surface 
which has been flattened in this manner. 

As for the other method of division along the plane 
GHKL, it will be seen that each spheroid would have to 
detach itself from four of the neighbouring layer, two of 
which touch it on the flattened surfaces, and two at the 
edges. So that this division is likewise more difficult than 
that which is made parallel to one of the surfaces of the 
crystal; where, as we have said, each spheroid is detached 
from only three of the neighbouring layer: of which three 
there is one only which touches it on the flattened surface, 
and the other two at the edges only. 

However, that which has made me know that in the 
crystal there are layers in this last fashion, is that in a piece 
weighing half a pound which I possess, one sees that it is 
split along its length, as is the above-mentioned prism by 
the plane GHKL; as appears by colours of the Iris ex- 
tending throughout this whole plane although the two 
pieces still hold together. All this proves then that the 
composition of the crystal is such as we have stated. To 



which I again add this experiment; that if one passes a 
knife scraping along any one of the natural surfaces, and 
downwards as it were from the equilateral obtuse angle, 
that is to say from the apex of the pyramid, one finds it 
quite hard; but by scraping in the opposite sense an inci- 
sion is easily made. This follows manifestly from the 
situation of the small spheroids; over which, in the first 
manner, the knife glides; but in the other manner it seizes 
them from beneath almost as if they were the scales of a 

I will not undertake to say anything touching the way 
in which so many corpuscles all equal and similar are 
generated, nor how they are set in such beautiful order; 
whether they are formed first and then assembled, or 
whether they arrange themselves thus in coming into being 
and as fast as they are produced, which seems to me more 
probable. To develop truths so recondite there would be 
needed a knowledge of nature much greater than that 
which we have. I will add only that these little spheroids 
could well contribute to form the spheroids of the waves 
of light, here above supposed, these as well as those being 
similarly situated, and with their axes parallel. 

Calculations which have been supposed in this Chapter. 

Mr. Bartholinus, in his treatise of this Crystal, puts at 
101 degrees the obtuse angles of the faces, which I have 
stated to be 101 degrees 52 minutes. He states that he 
measured these angles directly on the crystal, which is 
difficult to do with ultimate exa&itude, because the edges 
such as CA, CB, in this figure, are generally worn, and not 
quite straight. For more certainty, therefore, I preferred to 
measure actually the obtuse angle by which the faces 



CBDA, CBVF, are inclined to one another, namely the 
angle OCN formed by drawing CN perpendicular to FV, 

and CO perpendi- 
cular to DA. This 
angle OCN I found 
to be 105 degrees; 
and its supplement 
CNP, to be 75 de- 
M grees, as it should be. 
To find from this 
the obtuse angle 
BCA, I imagined a 
sphere having its centre at C, and on its surface a spherical 
triangle, formed by the intersection of three planes which 
enclose the solid angle C. In this equilateral triangle, 
which is ABF in this other figure, I see A 

that each of the angles should be 105 
degrees, namely equal to the angle OCN ; 
and that each of the sides should be of 
as many degrees as the angle ACB, or 
ACF, or BCF. Having then drawn the 
arc FQ perpendicular to the side AB, 
which it divides equally at Q, the triangle 
FQA has a right angle at Q, the angle A 105 degrees, and 
F half as much, namely 52 degrees 30 minutes; whence 
the hypotenuse AF is found to be 101 degrees 52 minutes. 
And this arc AF is the measure of the angle ACF in the 
figure of the crystal. 

In the same figure, if the plane CGHF cuts the crystal 
so that it divides the obtuse angles ACB, MHV, in the 
middle, it is stated, in Article 10, that the angle CFH is 
70 degrees 57 minutes. This again is easily shown in the 



same spherical triangle ABF, in which it appears that 
the arc FQ is as many degrees as the angle GCF in the 
crystal, the supplement of which is the angle CFH. Now 
the arc FQ is found to be 109 degrees 3 minutes. Then 
its supplement, 70 degrees 57 minutes, is the angle CFH. 

It was stated, in Article 26, that the straight line CS, 
which in the preceding figure is CH, being the axis of the 
crystal, that is to say being equally inclined to the three 
sides CA, CB, CF, the angle GCH is 45 degrees 20 
minutes. This is also easily calculated by the same spherical 
triangle. For by drawing the other arc AD which cuts 
BF equally, and intersects FQ at S, this point will be the 
centre of the triangle. And it is easy to see that the arc 
SQ is the measure of the angle GCH in the figure 
which represents the crystal. Now in the triangle QAS, 
which is right-angled, one knows also the angle A, which 
is 52 degrees 30 minutes, and the side AQ 50 degrees 56 
minutes; whence the side SQ is found to be 45 degrees 
20 minutes. 

In Article 27 it was required to show that PMS being 
an ellipse the centre of which is C, and which touches 
the straight line MD at M so that the angle MCL which 
CM makes with CL, perpendicular on DM, is 6 degrees 
40 minutes, and its semi-minor axis CS making with CG 
(which is parallel to MD) an angle GCS of 45 degrees 20 
minutes, it was required to show, I say, that, CM being 
100,000 parts, PC the semi-major diameter of this ellipse 
is 105,032 parts, and CS, the semi-minor diameter, 93,410. 

Let CP and CS be prolonged and meet the tangent 
DM at D and Z; and from the point of contadt M let 
MN and MO be drawn as perpendiculars to CP and CS. 
Now because the angles SCP, GCL, are right angles, the 



angle PCL will be equal to GCS which was 45 degrees 
20 minutes. And deducting the angle LCM, which is 6 

degrees 40 minutes, from 
LCP, which is 45 degrees 20 
minutes, there remains MCP, 
38 degrees 40 minutes. Con- 
sidering then CM as a radius 
of 100,000 parts, MN, the 
sine of 38 degrees 40 minutes, 
will be 62,479. And in the 
right-angled triangle MND, 
MN will be to ND as the 
radius of the Tables is to the 
tangent of 45 degrees 20 minutes (because the angle 
NMD is equal to DCL, or GCS); that is to say as 
100,000 to 101,170: whence results ND 63,210. But 
NC is 78,079 of the same parts, CM being 100,000, 
because NC is the sine of the complement of the angle 
MCP, which was 38 degrees 40 minutes. Then the 
whole line DC is 141,289; and CP, which is a mean 
proportional between DC and CN, since MD touches the 
Ellipse, will be 105,032. 

Similarly, because the angle OMZ is equal to CDZ, or 
LCZ, which is 44 degrees 40 minutes, being the com- 
plement of GCS, it follows that, as the radius of the 
Tables is to the tangent of 44 degrees 40 minutes, so will 
OM 78,079 be to OZ 77,176. But OC is 62,479 of these 
same parts of which CM is 100,000, because it is equal to 
MN, the sine of the angle MCP, which is 38 degrees 
40 minutes. Then the whole line CZ is 139,655; and 
CS, which is a mean proportional between CZ and CO 
will be 93,410. 



At the same place it was stated that GC was found to 
be 98,779 parts. To prove this, let PE be drawn in the 
same figure parallel to DM, and meeting CM at E. In 
the right-angled triangle CLD the side CL is 99,324 
(CM being 100,000), because CL is the sine of the com- 
plement of the angle LCM, which is 6 degrees 40 
minutes. And since the angle LCD is 45 degrees 20 
minutes, being equal to GCS, the side LD is found to be 
100,486 : whence deducing ML 1 1,609 there w ^l remain 
MD 88,877. Now as CD (which was 141,289) is to DM 
88,877, so will CP 105,032 be to PE 66,070. But as the 
reftangle MEH (or rather the difference of the squares 
on CM and CE) is to the square on MC, so is the square 
on PE to the square on Cg-; then also as the difference of 
the squares on DC and CP to the square on CD, so also 
is the square on PE to the square on gC. But DP, CP, 
and PE are known ; hence also one knows GC, which is 

Lemma which has been supposed. 

If a spheroid is touched by a straight line, and also by 
two or more planes which are parallel to this line, though 
not parallel to one another, all the points of conta6t of 
the line, as well as of the planes, will be in one and the 
same ellipse made by a plane which passes through the 
centre of the spheroid. 

Let LED be the spheroid touched by the line BM at 
the point B, and also by the planes parallel to this line at 
the points O and A. It is required to demonstrate that 
the points B, O, and A are in one and the same Ellipse 
made in the spheroid by a plane which passes through 
its centre. 



Through the line BM, and through the points O and 
A, let there be drawn planes parallel to one another, 

which, in cutting the spher- 
oid make the ellipses LED, 
POP,QAQ; which will all 
be similar and similarly dis- 
posed, and will have their 
centres K, N, R, in one and 
the same diameter of the 
spheroid, which will also be 
the diameter of the ellipse 
made by the section of the 
plane that passes through the 
centre of the spheroid, and 
which cuts the planes of the 
three said Ellipses at right 
angles: for all this is manifest by proposition 15 of the 
book of Conoids and Spheroids of Archimedes. Further, 
the two latter planes, which are drawn through the points 
O and A, will also, by cutting the planes which touch the 
spheroid in these same points, generate straight lines, as OH 
and AS, which will, as is easy to see, be parallel to BM ; 
and all three, BM, OH, AS, will touch the Ellipses LED, 
POP, QAQ in these points, B, O, A ; since they are in 
the planes of these ellipses, and at the same time in the 
planes which touch the spheroid. If now from these 
points B, O, A, there are drawn the straight lines BK, 
ON, AR, through the centres of the same ellipses, and 
if through these centres there are drawn also the diameters 
LD, PP, QQ, parallel to the tangents BM, OH, AS; 
these will be conjugate to the aforesaid BK, ON, AR. 
And because the three ellipses are similar and similarly 



disposed, and have their diameters LD, PP, QQ parallel, 
it is certain that their conjugate diameters BK, ON, AR, 
will also be parallel. And the centres K, N, R being, as 
has been stated, in one and the same diameter of the 
spheroid, these parallels BK, ON, AR will necessarily 
be in one and the same plane, which passes through this 
diameter of the spheroid, and, in consequence, the points 
R, O, A are in one and the same ellipse made by the in- 
terse6tion of this plane. Which was to be proved. And 
it is manifest that the demonstration would be the same 
if, besides the points O, A, there had been others in which 
the spheroid had been touched by planes parallel to the 
straight line BM. 


Which serve for Refraction and for Reflexion. 

FTER having explained how the properties 
of reflexion and refraction follow from 
what we have supposed concerning the 
nature of light, and of opaque bodies, and 
of transparent media, I will here set forth 
a very easy and natural way of deducing, 

from the same principles, the true figures which serve, 
either by reflexion or by refradtion, to colleft or disperse 
the rays of light, as may be desired. For though I do not 
see yet that there are means of making use of these figures, 
so far as relates to Refraction, not only because of the 
difficulty of shaping the glasses of Telescopes with the re- 

p quisite 



quisite exa6litude according to these figures, but also be- 
cause there exists in refra&ion itself a property which 
hinders the perfeft concurrence of the rays, as Mr. Newton 
has very well proved by experiment, I will yet not desist 
from relating the invention, since it offers itself, so to speak, 
of itself, and because it further confirms our Theory of re- 
fraftion, by the agreement which here is found between 
the refrafted ray and the reflefted ray. Besides, it may 
occur that some one in the future will discover in it utili- 
ties which at present are not seen. 

To proceed then to these figures, let us suppose first 
that it is desired to find a surface CDE which shall re- 
assemble at a point B rays coming from another point A ; 
and that the summit of the surface shall be the given 
point D in the straight line AB. I say that, whether by 

reflexion or by refradlion, it is only necessary to make 
this surface such that the path of the light from the point 
A to all points of the curved line CDE, and from these to 
the point of concurrence (as here the path along the straight 
lines AC, CB, along AL, LB, and along AD, DB), shall 
be everywhere traversed in equal times : by which prin- 
ciple the finding of these curves becomes very easy. 



So far as relates to the reflecting surface, since the sum 
of the lines AC, CB ought to be equal to that of AD, DB, 
it appears that DCE ought 
to be an ellipse; and for 
refraction, the ratio of the 
velocities of waves of light 
in the media A and B 
being supposed to be 
known, for example that 
of 3 to 2 (which is the 
same, as we have shown, 
as the ratio of the Sines 
in the refraction), it is only necessary to make DH equal 
to -f of DB; and having after that described from the centre 
A some arc FC, cutting DB at F, then describe another 
from centre B with its semi-diameter BX equal to -| of 
FH ; and the point of intersection of the two arcs will be 
one of the points required, through which the curve should 
pass. For this point, having been found in this fashion, it is 
easy forthwith to demonstrate that the time along AC, 
CB, will be equal to the time along AD, DB. 

For assuming that the line AD represents the time which 
the light takes to traverse this same distance AD in air, 
it is evident that DH, equal to f of DB, will represent the 
time of the light along DB in the medium, because it needs 
here more time in proportion as its speed is slower. There- 
fore the whole line AH will represent the time along 
AD, DB. Similarly the line AC or AF will represent the 
time along AC; and FH being by construction equal 
to f of CB, it will represent the time along CB in the me- 
dium; and in consequence the whole line AH will represent 
also the time along AC, CB. Whence it appears that the 



time along AC, CB, is equal to the time along AD, DB. 
And similarly it can be shown if L and K are other points 
in the curve CDE, that the times along AL, LB, and along 
AK, KB, are always represented by the line AH, and 
therefore equal to the said time along AD, DB. 

In order to show further that the surfaces, which these 
curves will generate by revolution, will dire6l all the rays 
which reach them from the point A in such wise that they 
tend towards B, let there be supposed a point K in the 
curve, farther from D than C is, but such that the straight 
line AK falls from outside upon the curve which serves for 
the refraction; ancj from the centre B let the arc KS be de- 
scribed, cutting BD at S, and the straight line CB at R; and 
from the centre A describe the arc DN meeting AK at N. 

Since the sums of the times along AK, KB, and along 
AC, CB are equal, if from the former sum one deducts 
the time along KB, and if from the other one deducts the 
time along RB, there will remain the time along AK as 
equal to the time along the two parts AC, CR. Conse- 
quently in the time that the light has come along AK it will 
also have come along AC and will in addition have made, 
in the medium from the centre C, a partial spherical 
wave, having a semi-diameter equal to CR. And this 
wave will necessarily touch the circumference KS at R, 
since CB cuts this circumference at right angles. Simi- 
larly, having taken any other point L in the curve, one 
can show that in the same time as the light passes along 
AL it will also have come along AL and in addition will 
have made a partial wave, from the centre L, which will 
touch the same circumference KS. And so with all other 
points of the curve CDE. Then at the moment that the 
light reaches K the arc KRS will be the termination 



of the movement, which has spread from A through 
DCK. And thus this same arc will constitute in the 
medium the propagation of the wave emanating from A ; 
which wave may be represented by the arc DN, or by 
any other nearer the centre A. But all the pieces of the 
arc KRS are propagated successively along straight lines 
which are perpendicular to them, that is to say, which 
tend to the centre B (for that can be demonstrated in the 
same way as we have proved above that the pieces of 
spherical waves are propagated along the straight lines 
coming from their centre), and these progressions of the 
pieces of the waves constitute the rays themselves of light. 
It appears then that all these rays tend here towards the 
point B. 

One might also determine the point C, and all the 
others, in this curve which serves for the refraftion, by 
dividing DA at G in such a way that DG is -| of DA, 
and describing from the centre B any arc CX which cuts 
BD at X, and another from the centre A with its semi- 
diameter AF equal to f of GX ; or rather, having de- 
scribed, as before, the arc CX, it is only necessary to make 
DF equal to f of DX, and from he centre A to strike the 
arc FC ; for these two constructions, as may be easily 
known, come back to the first one which was shown 
before. And it is manifest by the last method that this 
curve is the same that Mr. Des Cartes has given in his 
Geometry, and which he calls the first of his Ovals. 

It is only a part of this oval which serves for the 
refraction, namely, the part DK, ending at K, if AK is 
the tangent. As to the other part, Des Cartes has re- 
marked that it could serve for reflexions, if there were 
some material of a mirror of such a nature that by its 



means the force of the rays (or, as we should say, the 
velocity of the light, which he could not say, since he 
held that the movement of light was instantaneous) 
could be augmented in the proportion of 3 to 2. But we 
have shown that in our way of explaining reflexion, such 


a thing could not arise from the matter of the mirror, and 
it is entirely impossible. 

From what has been demonstrated about this oval, it 
will be easy to find the figure which serves to collect to a 
point incident parallel rays. For by supposing just the 
same construction, but the point A infinitely distant, 
giving parallel rays, our oval becomes a true Ellipse, the 



construction of which differs in no way from that of the 
oval, except that FC, which previously was an arc of a 
circle, is here a straight line, perpendicular to DB. For 
the wave of light DN, being likewise represented by a 
straight line, it will be seen that all the points of this 
wave, travelling as far as the surface KD along lines 
parallel to DB, will advance subsequently towards the 
point B 5 and will arrive there at the same time. As for 
the Ellipse which served for reflexion, it is evident that 
it will here become a parabola, since its focus A may be 
regarded as infinitely distant from the other, B, which is 
here the focus of the parabola, towards which all the 
reflexions of rays parallel to AB tend. And the demon- 
stration of these effects is just the same as the preceding. 

But that this curved line CDE which serves for refrac- 
tion is an Ellipse, and is such that its major diameter is to 
the distance between its foci as 3 to 2, which is the 
proportion of the refra6lion, can be easily found by the 
calculus of Algebra. For DB, which is given, being called 
a ; its undetermined perpendicular DT being called x ; and 
TC y; FB will be ay\ CB will be \/xx + aa2ay+yy. 
But the nature of the curve is such that J- of TC together 
with CB is equal to DB, as was stated in the last construction : 
then the equation will be between \y + ^/xx + aa 2 ay+yy 
and a ; which being reduced, gives \ay-yy equal to-f-x*; 
that is to say that having made DO equal to of DB, the 
rectangle DFO is equal to |- of the square on FC. Whence 
it is seen that DC is an ellipse, of which the axis DO is to 
the parameter as 9 to 5 ; and therefore the square on DO 
is to the square of the distance between the foci as 9 to 
9 5, that is to say 4; and finally the line DO will be to 
this distance as 3 to 2. 


I 12 


Again, if one supposes the point B to be infinitely dis- 
tant, in lieu of our first oval we shall find that CDE is a true 
Hyperbola; which will make those rays become parallel 
which come from the point A. And in consequence also 
those which are parallel within the transparent body will 
be collected outside at the point A. Now it must be re- 
marked that CX and KS become straight lines perpendi- 
cular to BA, because they represent arcs of circles the 

centre of which is infinitely distant. And the intersection 
of the perpendicular CX with the arc FC will give the 
point C, one of those through which the curve ought to 
pass. And this operates so that all the parts of the wave 
of light DN, coming to meet the surface KDE, will 
advance thence along parallels to KS and will arrive at 
this straight line at the same time; of which the proof is 
again the same as that which served for the first oval. 
Besides one finds by a calculation as easy as the preceding 
one, that CDE is here a hyperbola of which the axis DO 



is -- of AD, and the parameter equal to AD. Whence it 
is easily proved that DO is to the distance between the 
foci as 3 to 2. 

These are the two cases in which Conic se6lions serve 
for refra6tion, and are the same which are explained, in 
his Dioptrique, by Des Cartes, who first found out the use 
of these lines in relation to refra&ion, as also that of the 


Ovals the first of which we have already set forth. The 
second oval is that which serves for rays that tend to a 
given point; in which oval, if the apex of the surface 
which receives the rays is D, it will happen that the other 
apex will be situated between B and A, or beyond A, 
according as the ratio of AD to DB is given of greater or 
lesser value. And in this latter case it is the same as that 
which Des Cartes calls his 3rd oval. 

Now the finding and construction of this second oval is 

Q the 


the same as that of the first, and the demonstration of its 
effedt likewise. But it is worthy of remark that in one 

case this oval be- 
comes a perfedt 
circle, namely 
when the ratio of 
AD to DB is the 
same as the ratio 
of the refractions, 
here as 3 to 2, as 
I observed a long 
A time ago. The 4th 
oval, serving only 
for impossible re- 
flexions, there is 
no need to set it 

As for the manner in which Mr. Des Cartes discovered 
these lines, since he has given no explanation of it, nor any 
one else since that I know of, I will say here, in passing, 
what it seems to me it must have been. Let it be proposed 
to find the surface generated by the revolution of the curve 
KDE, which, receiving the incident rays coming to it from 
the point A, shall deviate them toward the point B. Then 
considering this other curve as already known, and that its 
apex D is in the straight line AB, let us divide it up into 
an infinitude of small pieces by the points G, C, F; and 
having drawn from each of these points, straight lines to- 
wards A to represent the incident rays, and other straight 
lines towards B, let there also be described with centre A 
the arcs GL, CM, FN, DO, cutting the rays that come 
from A at L, M, N, O; and from the points K, G, C, F, 



let there be described the arcs KQ, GR, CS, FT cutting 
the rays towards B at Q, R, S, T; and let us suppose that 
the straight line HKZ cuts the curve at K at right-angles. 
Then AK being an incident ray, and KB its refraftion 
within the medium, it needs must be, according to the law 

of refraftion which was known to Mr. Des Cartes, that 
the sine of the angle ZKA should be to the sine of the 
angle HKB as 3 to 2, supposing that this is the proportion 
of the refraftion of glass; or rather, that the sine of the 
angle KGL should have this same ratio to the sine of the 
angle GKQ, considering KG, GL, KQ as straight lines 
because of their smallness. But these sines are the lines 
KL and GQ, if GK is taken as the radius of the circle. 
Then LK ought to be to GQ as 3 to 2; and in the same 
ratio MG to CR, NC to FS, OF to DT. Then also the 
sum of all the antecedents to all the consequents would be 
as 3 to 2. Now by prolonging the arc DO until it meets 
AK at X, KX is the sum of the antecedents. And by 
prolonging the arc KQ till it meets AD at Y, the sum of 



the consequents is DY. Then KX ought to be to DY as 
3 to 2. Whence it would appear that the curve KDE was 
of such a nature that having drawn from some point which 
had been assumed, such as K, the straight lines KA, KB, the 
excess by which AK surpasses AD should be to the excess 
of DB over KB, as 3 to 2. For it can similarly be demon- 
strated, by taking any other point in the curve, such as G, 
that the excess of AG over AD, namely VG, is to the 
excess of BD over DG, namely DP, in this same ratio of 
3 to 2. And following this principle Mr. Des Cartes 
constructed these curves in his Geometric; and he easily 
recognized that in the case of parallel rays, these curves 
became Hyperbolas and Ellipses. 

Let us now return to our method and let us see how it 
leads without difficulty to the finding of the curves which 
one side of the glass requires when the other side is of a 
given figure; a figure not only plane or spherical, or made 
by one of the conic sections (which is the restriction with 
which Des Cartes proposed this problem, leaving the solu- 
tion to those who should come after him) but generally 
any figure whatever: that is to say, one made by the 
revolution of any given curved line to which one must 
merely know how to draw straight lines as tangents. 

Let the given figure be that made by the revolution of 
some curve such as AK about the axis AV, and that this 
side of the glass receives rays coming from the point L. 
Furthermore, let the thickness AB of the middle of the 
glass be given, and the point F at which one desires the 
rays to be all perfectly reunited, whatever be the first re- 
fra6lion occurring at the surface AK. 

I say that for this the sole requirement is that the out- 
line BDK which constitutes the other surface shall be 




such that the path 
of the light from 
the point L to the 
surface AK, and 
from thence to the 
surface BDK, and 
from thence to the 
point F, shall be 
traversed every- 
where in equal 
times, and in each 
case in a time 
equal to that which 
the light employs K/ 
to pass along the 
straight line LF 
of which the part 
AB is within the 

Let LG be a ray 
falling on the arc 
AK. Its refraHon 
GV will be given 
by means of the 
tangent which will 
be drawn at the 
point G. Now in 
GV the point D 
must be found such 
that FD together 
with f of DG and 
the straight line 




GL, may be equal to FB together with f of BA and 
the straight line AL ; which, as is clear, make up a given 
length. Or rather, by deducing from each the length of 
LG, which is also given, it will merely be needful to adjust 
FD up to the straight line VG in such a way that FD 
together with -f of DG is equal to a given straight line, 
which is a quite easy plane problem: and the point D 
will be one of those through which the curve BDK ought 
to pass. And similarly, having drawn another ray LM, 
and found its refraction MO, the point N will be found in 
this line, and so on as many times as one desires. 

To demonstrate the effe6t of the curve, let there be 
described about the centre L the circular arc AH, cutting 
LG at H ; and about the centre F the arc BP; and in AB 
let AS be taken equal to | of HG ; and SE equal to GD. 
Then considering AH as a wave of light emanating from 
the point L, it is certain that during the time in which its 
piece H arrives at G the piece A will have advanced within 
the transparent body only along AS ; for I suppose, as above, 
the proportion of the refra&ion to be as 3 to 2. Now we 
know that the piece of wave which is incident on G, ad- 
vances thence along the line GD, since GV is the refradtion 
of the ray LG. Then during the time that this piece of 
wave has taken from G to D, the other piece which was 
at S has reached E, since GD, SE are equal. But while 
the latter will advance from E to B, the piece of wave 
which was at D will have spread into the air its partial 
wave, the semi-diameter of which, DC (supposing this 
wave to cut the line DF at C), will be f of EB, since the 
velocity of light outside the medium is to that inside as 3 
to 2. Now it is easy to show that this wave will touch the 
arc BP at this point C. For since, by construction, FD + 



I- DG + GL are equal to FB + | BA + AL ; on dedud- 
ing the equals LH, LA, there will remain FD + fDG + 
GH equal to FB + f BA. And, again, deducting from 

one side GH, and from the other side -f- of AS, which are 


equal, there will remain FD with -f DG equal to FB with 
! of BS. But f of DG are equal to f of ES ; then FD is 
equal to FB with | of BE. But DC was equal to f of EB ; 
then deducing these equal lengths from one side and from 
the other, there will remain CF equal to FB. And thus 
it appears that the wave, the semi-diameter of which is DC, 
touches the arc BP at the moment when the light coming 
from the point L has arrived at B along the line LB. It 
can be demonstrated similarly that at this same moment 
the light that has come along any other ray, such as LM, 
MN, will have propagated the movement which is termi- 
nated at the arc BP. Whence it follows, as has been often 
said, that the propagation of the wave AH, after it has passed 
through the thickness of the glass, will be the spherical 
wave BP, all the pieces of which ought to advance along 
straight lines, which are the rays of light, to the centre F. 
Which was to be proved. Similarly these curved lines can 
be found in all the cases which can be proposed, as will be 
sufficiently shown by one or two examples which I will add. 

Let there be given the surface of the glass AK, made 
by the revolution about the axis BA of the line AK, which 
may be straight or curved. Let there be also given in the 
axis the point L and the thickness BA of the glass; and let 
it be required to find the other surface KDB, which receiv- 
ing rays that are parallel to AB will dire<5t them in such 
wise that after being again refracted at the given surface 
AK they will all be reassembled at the point L. 

From the point L let there be drawn to some point of 



the given line AK the straight line LG, which, being 
considered as a ray of light, its refraction GD will then 

be found. And this line 
being then prolonged at 
one side or the other will 
meet the straight line BL, 
as here at V. Let there 
then be erefted on AB the 
perpendicular BC, which 
will represent a wave of 
light coming from the in- 
finitely distant point F, 
since we have supposed 
the rays to be parallel. 
Then all the parts of this 
wave BC must arrive at 
the same time at the 
point L; or rather all the 
parts of a wave emanating 
from the point L must 
arrive at the same time at the straight line BC. And for 
that, it is necessary to find in the line VGD the point D 
such that having drawn DC parallel to AB, the sum of 
CD, plus of DG, plus GL may be equal to | of AB, 
plus AL : or rather, on deducting from both sides GL, 
which is given, CD plus f of DG must be equal to a 
given length ; which is a still easier problem than the 
preceding construction. The point D thus found will be 
one of those through which the curve ought to pass ; and 
the proof will be the same as before. And by this it will 
be proved that the waves which come from the point L, 
after having passed through the glass KAKB, will take 



the form of straight lines, as BC ; which is the same 
thing as saying that the rays will become parallel. Whence 
it follows reciprocally that paral- 
lel rays falling on the surface 
KDB will be reassembled at the 
point L. 

Again, let there be given the 
surface AK, of any desired form, 
generated by revolution about 
the axis AB, and let the thick- 
ness of the glass at the middle 
be AB. Also let the point L 
be given in the axis behind the 
glass; and let it be supposed 
that the rays which fall on the 
surface AK tend to this point, 
and that it is required to find 
the surface BD, which on their 
emergence from the glass turns them as if they came 
from the point F in front of the glass. 

Having taken any point G in the line AK, and drawing 
the straight line IGL, its part GI will represent one of 
the incident rays, the refraction of which, GV, will then 
be found : and it is in this line that we must find the 
point D, one of those through which the curve DG ought 
to pass. Let us suppose that it has been found : and about 
L as centre let there be described GT, the arc of a circle 
cutting the straight line AB at T, in case the distance LG 
is greater than LA ; for otherwise the arc AH must be 
described about the same centre, cutting the straight line 
LG at H. This arc GT (or AH, in the other case) will 
represent an incident wave of light, the rays of which 

R tend 


tend towards L. Similarly, about the centre F let there 
be described the circular arc DQ, which will represent a 
wave emanating from the point F. 

Then the wave TG, after having passed through the 
glass, must form the wave QD ; and for this I observe 
that the time taken by the light along GD in the glass 
must be equal to that taken along the three, TA, AB, 
and BQ, of which AB alone is within the glass. Or 
rather, having taken AS equal to f of AT, I observe that 
| of GD ought to be equal to J~ of SB, plus BQ ; and, 
deducting both of them from FD or FQ, that FD less 
| of GD ought to be equal to FB less | of SB. And 
this last difference is a given length : and all that is 
required is to draw the straight line FD from the given 
point F to meet VG so that it may be thus. Which is a 
problem quite similar to that which served for the first 
of these constructions, where FD plus f of GD had to 
be equal to a given length. 

In the demonstration it is to be observed that, since 
the arc BC falls within the glass, there must be conceived 
an arc RX, concentric with it and on the other side of QD. 
Then after it sjiall have been shown that the piece G of 
the wave GT arrives at D at the same time that the piece 
T arrives at Q, which is easily deduced from the construc- 
tion, it will be evident as a consequence that the partial 
wave generated at the point D will touch the arc RX at 
the moment when the piece Q shall have come to R, and 
that thus this arc will at the same moment be the termi- 
nation of the movement that comes from the wave TG ; 
whence all the rest may be concluded. 

Having shown the method of finding these curved 
lines which serve for the perfect concurrence of the rays, 



there remains to be explained a notable thing touching 
the uncoordinated refraction of spherical, plane, and other 
surfaces : an effeCt which if ignored might cause some 
doubt concerning what we have several times said, that 
rays of light are 
straight lines which 
intersect at right D 
angles the waves which 
travel along them. 



For in the case 
rays which, for ex- 
ample, fall parallel 
upon a spherical sur- 
face AFE, intersecting 
one another, after re- 
fraCtion, at different 
points, as this figure 
represents; what can 
the waves of light 
be, in this transparent 
body, which are cut 
at right angles by the 
converging rays? For 
they oan not be spheri- 
cal. And what will 
these waves become 
after the said rays begin to intersect one another ? It 
will be seen in the solution of this difficulty that some- 
thing very remarkable comes to pass herein, and that the 
waves do not cease to persist though they do not continue 
entire, as when they cross the glasses designed according 
to the construction we have seen. 



According to what has been shown above, the straight 
line AD, which has been drawn at the summit of the 
sphere, at right angles to the axis parallel to which the 
rays come, represents the wave of light; and in the time 
taken by its piece D to reach the spherical surface AGE 
at E, its other parts will have met the same surface at 
F, G, H, etc., and will have also formed spherical partial 
waves of which these points are the centres. And the 
surface EK which all those waves will touch, will be the 
continuation of the wave AD in the sphere at the moment 
when the piece D has reached E. Now the line EK is 
not an arc of a circle, but is a curved line formed as the 
evolute of another curve ENC, which touches all the 
rays HL, GM, FO, etc., that are the refraftions of the 
parallel rays, if we imagine laid over the convexity ENC 
a thread which in unwinding describes at its end E the 
said curve EK. For, supposing that this curve has been 
thus described, we will show that the said waves formed 
from the centres F, G, H, etc., will all touch it. 

It is certain that the curve EK and all the others described 
by the evolution of the curve ENC, with different lengths 
of thread, will cut all the rays HL, GM, FO, etc., at right 
angles, and in such wise that the parts of them intercepted 
between two such curves will all be equal ; for this follows 
from what has been demonstrated in our treatise de Motu 
Pendulorum. Now imagining the incident rays as being 
infinitely near to one another, if we consider two of them, 
as RG, TF, and draw GQ perpendicular to RG, and if we 
suppose the curve FS which intersects GM at P to have 
been described by evolution from the curve NC, beginning 
at F,as far as which the thread is supposed to extend, we may 
assume the small piece FP as a straight line perpendicular 



to the ray GM, and similarly the arc GF as a straight 
line. But GM being the refradion of the ray RG, and FP 
being perpendicular to it, QF must be to GP as 3 to 2, that 
is to say in the proportion of the refra&ion ; as was shown 
above in explaining the discovery of Des Cartes. And the 
same thing occurs in all the small arcs GH, HA, etc., 
namely that in the quadrilaterals which enclose them the 
side parallel to the axis is to the opposite side as 3 to 2. 
Then also as 3 to 2 will the sum of the one set be to 
the sum of the other; that is to say, TF to AS, and DE 
to AK, and BE to SK or DV, supposing V to be the inter- 
section of the curve EK and the ray FO. But, making FB 
perpendicular to DE, the ratio of 3 to 2 is also that of BE 
to the semi-diameter of the spherical wave which emanated 
from the point F while the light outside the transparent 
body traversed the space BE. Then it appears that this 
wave will intersedl the ray FM at the same point V where 
it is interse6led at right angles by the curve EK, and con- 
sequently that the wave will touch this curve. In the same 
way it can be proved that the same will apply to all the 
other waves above mentioned, originating at the points G, 
H, etc.; to wit, that they will touch the curve EK at the 
moment when the piece D of the wave ED shall have 
reached E. 

Now to say what these waves become after the rays have 
begun to cross one another: it is that from thence they 
fold back and are composed of two contiguous parts, 
one being a curve formed as evolute of the curve ENC 
in one sense, and the other as evolute of the same curve 
in the opposite sense. Thus the wave KE, while advanc- 
ing toward the meeting place becomes abc^ whereof the 
part ab is made by the evolute ^C, a portion of the curve 



ENC, while the end C remains attached; and the part be 
by the evolute of the portion E while the end E remains 
attached. Consequently the same wave becomes def^ then 
ghkj and finally CY, from whence it subsequently spreads 
without any fold, but always along curved lines which are 
evolutes of the curve ENC, increased by some straight 
line at the end C. 

There is even, in this curve, a part EN which is straight, 
N being the point where the perpendicular from the centre 
X of the sphere falls upon the refraCtion of the ray DE, 
which I now suppose to touch the sphere. The folding 
of the waves of light begins from the point N up to the 
end of the curve C, which point is formed by taking AC 
to CX in the proportion of the refraction, as here 3 to 2. 

As many other points as may be desired in the curve 
NC are found by a Theorem which Mr. Barrow has de- 
monstrated in section 1 2 of his LeStiones Opticae, though 
for another purpose. And it is to be noted that a straight 
line equal in length to this curve can be given. For since 
it together with the line NE is equal to the line CK, 
which is known, since DE is to AK in the proportion of 
the refraction, it appears that by deducting EN from CK 
the remainder will be equal to the curve NC. 

Similarly the waves that are folded back in reflexion by 
a concave spherical mirror can be found. Let ABC be the 
sedtion, through the axis, of a hollow hemisphere, the 
centre of which is D, its axis being DB, parallel to which 
I suppose the rays of light to come. All the reflexions of 
those rays which fall upon the quarter-circle AB will 
touch a curved line AFE, of which line the end E is at 
the focus of the hemisphere, that is to say, at the point 
which divides the semi-diameter BD into two equal parts. 



The points through which this curve ought to pass are 

found by taking, beyond A, some arc AO, and making 

the arc OP double 

the length of it ; then 

dividing the chord 

OP at F in such 

wise that the part 

FP is three times the 

part FO; for then F 

is one of the required A I 


And as the parallel 
rays are merely per- 
pendiculars to the 
waves which fall on the concave surface, which waves are 
parallel to AD, it will be found that as they come succes- 
sively to encounter the surface AB, they form on reflexion 
folded waves composed of two curves which originate from 
two opposite evolutions of the parts of the curve AFE. So, 
taking AD as an incident wave, when the part AG shall 
have met the surface AI, that is to say when the piece G 
shall have reached I, it will be the curves HF, FI, gener- 
ated as evolutes of the curves FA, FE, both beginning at 
F, which together constitute the propagation of the part 
AG. And a little afterwards, when the part AK has met 
the surface AM, the piece K having come to M, then the 
curves LN, NM, will together constitute the propagation 
of that part. And thus this folded wave will continue to 
advance until the point N has reached the focus E. The 
curve AFE can be seen in smoke, or in flying dust, when a 
concave mirror is held opposite the sun. And it should be 
known that it is none other than that curve which is de- 


scribed by the point E on the circumference of the circle 
EB, when that circle is made to roll within another whose 
semi-diameter is ED and whose centre is D. So that it is 
a kind of Cycloid, of which, however, the points can be 
found geometrically. 

Its length is exactly equal to J of the diameter of the 
sphere, as can be found and demonstrated by means of 
these waves, nearly in the same way as the mensuration of 
the preceding curve; though it may also be demonstrated 
in other ways, which I omit as outside the subjedt. The 
area AOBEFA, comprised between the arc of the quarter- 
circle, the straight line BE, and the curve EFA, is equal 
to the fourth part of the quadrant DAB.